1. Operations with Real Numbers

1.5. Exponents and Scientific Notation

Learning Objectives

By the end of this section it is expected that you will be able to:

  • Simplify expressions with exponents
  • Simplify expressions with zero exponents
  • Use the definition of a negative exponent
  • Use formulas with exponents in applications
  • Convert from decimal notation to scientific notation
  • Convert scientific notation to decimal form
  • Multiply and divide using scientific notation

Simplify Expressions with Exponents

Remember that an exponent indicates repeated multiplication of the same quantity. For example, {2}^{4} means to multiply 2 by itself 4 times, so {2}^{4} means 2 · 2 · 2 · 2

Let’s review the vocabulary for expressions with exponents.

Exponential Notation (Power)

This figure has two columns. In the left column is a to the m power. The m is labeled in blue as an exponent. The a is labeled in red as the base. In the right column is the text “a to the m power means multiply m factors of a.” Below this is a to the m power equals a times a times a times a, followed by an ellipsis, with “m factors” written below in blue.

This is read a to the {m}^{th} power.

In the expression {a}^{m}, the exponent m tells us how many times we use the base a as a factor.

This figure has two columns. The left column contains 4 cubed. Below this is 4 times 4 times 4, with “3 factors” written below in blue. The right column contains negative 9 to the fifth power. Below this is negative 9 times negative 9 times negative 9 times negative 9 times negative 9, with “5 factors” written below in blue.

Before we begin working with expressions containing exponents, let’s simplify a few expressions involving only numbers.

EXAMPLE 1

Simplify: a) {4}^{3} b) {7}^{1} c) {\left(\dfrac{5}{6}\right)}^{2} d) {\left(0.63\right)}^{2}.

Solution
a) {4}^{3}
Multiply three factors of 4. 4 · 4 · 4
Simplify. 64
b) {7}^{1}
Multiply one factor of 7. 7
c) {\left(\dfrac{5}{6}\right)}^{2}
Multiply two factors. \left(\dfrac{5}{6}\right)\left(\dfrac{5}{6}\right)
Simplify. \dfrac{25}{36}
d) {\left(0.63\right)}^{2}
Multiply two factors. \left(0.63\right)\left(0.63\right)
Simplify. 0.3969

TRY IT 1

Simplify: a) {6}^{3} b) {15}^{1} c) {\left(\dfrac{3}{7}\right)}^{2} d) {\left(0.43\right)}^{2}.

Show answer

a) 216 b) 15 c) \dfrac{9}{49} d) 0.1849

EXAMPLE 2

Simplify: a) {\left(-5\right)}^{4} b) -{5}^{4}.

Solution
a) {\left(-5\right)}^{4}
Multiply four factors of -5. \left(-5\right)\left(-5\right)\left(-5\right)\left(-5\right)
Simplify. 625
b) -{5}^{4}
Multiply four factors of 5. -(5  · 5 · 5 · 5)
Simplify. -625

TRY IT 2

Simplify: a) {\left(-3\right)}^{4} b) -{3}^{4}.

Show answer

a) 81 b) -81

Notice the similarities and differences in (Example 2) a) and (Example 2) b)! Why are the answers different? As we follow the order of operations in part a) the parentheses tell us to raise the \left(-5\right) to the 4th power. In part b) we raise just the 5 to the 4th power and then take the opposite.

 When simplifying with exponents instead of multiplying the same factors, we can use scientific calculator and a key labelled or  .           
For example, to find {\left(0.7\right)}^{5}, press: 0.7 5  image .You should get 0.16807.

Simplify Expressions with an Exponent of Zero

When simplifying expressions with exponents we very often use the Product Property and the Quotient Property.

Product Property for Exponents

If a is a real number, and m and n are counting numbers, then

{a}^{m}\cdot {a}^{n}={a}^{m+n}

To multiply with like bases, add the exponents.

An example with numbers helps to verify this property.

\begin{array}{rcl} {2}^{2}\cdot {2}^{3}& \stackrel{?}{=} & {2}^{2+3} \\ 4\cdot 8 & \stackrel{?}{=} & {2}^{5}\hfill \\ 32& = & 32\checkmark \hfill \end{array}

Quotient Property for Exponents

If a is a real number, a\ne 0, and m and n are whole numbers, then

\dfrac{{a}^{m}}{{a}^{n}}={a}^{m-n},m > n and \dfrac{{a}^{m}}{{a}^{n}}=\dfrac{1}{{a}^{n-m}},n > m

A couple of examples with numbers may help to verify this property.

\begin{array}{rlrl}\dfrac{{3}^{4}}{{3}^{2}}& = {3}^{4-2}\hfill & \hfill \dfrac{{5}^{2}}{{5}^{3}} &= \dfrac{1}{{5}^{3-2}} \\ \dfrac{81}{9} &= {3}^{2}\hfill & \dfrac{25}{125}& = \dfrac{1}{{5}^{1}} \\ 9& = 9\checkmark\hfill & \dfrac{1}{5}& = \dfrac{1}{5}\checkmark \end{array}

A special case of the Quotient Property is when the exponents of the numerator and denominator are equal, such as an expression like \dfrac{{a}^{m}}{{a}^{m}}. From your earlier work with fractions, you know that:

\dfrac{2}{2}=1 \hspace{1cm} \dfrac{17}{17}=1 \hspace{1cm} \dfrac{-43}{-43}=1

In words, a number divided by itself is 1. So, \dfrac{x}{x}=1, for any x\left(x\ne 0\right), since any number divided by itself is 1

The Quotient Property for Exponents shows us how to simplify \dfrac{{a}^{m}}{{a}^{n}} when m > n and when n < m by subtracting exponents. What if m=n?

Consider \frac{8}{8}, which we know is 1

\dfrac{8}{8}=1
Write 8 as {2}^{3}. \dfrac{{2}^{3}}{{2}^{3}}=1
Subtract exponents. {2}^{3-3}=1
Simplify. {2}^{0}=1

Now we will simplify \dfrac{{a}^{m}}{{a}^{m}} in two ways to lead us to the definition of the zero exponent. In general, for a\ne 0:

This figure is divided into two columns. At the top of the figure, the left and right columns both contain a to the m power divided by a to the m power. In the next row, the left column contains a to the m minus m power. The right column contains the fraction m factors of a divided by m factors of a, represented in the numerator and denominator by a times a followed by an ellipsis. All the as in the numerator and denominator are canceled out. In the bottom row, the left column contains a to the zero power. The right column contains 1.

We see \dfrac{{a}^{m}}{{a}^{m}} simplifies to {a}^{0} and to 1. So {a}^{0}=1.

Zero Exponent

If a is a non-zero number, then {a}^{0}=1.

Any nonzero number raised to the zero power is 1

EXAMPLE 3

Simplify:  {-9}^{0} .

Solution

The definition says any non-zero number raised to the zero power is 1

Use the definition of the zero exponent. \begin{array}{c}{9}^{0}\\ 1\end{array}

TRY IT 3

Simplify:  {-1.5}^{0}

Show answer

 1

Now that we have defined the zero exponent, we can expand all the Properties of Exponents to include whole number exponents.

Use the Definition of a Negative Exponent

 Now, let’s use the definition of a negative exponent to simplify expressions.

Negative Exponent

If n is an integer and a\ne 0, then {a}^{-n}=\frac{1}{{a}^{n}}.

The negative exponent tells us we can re-write the expression by taking the reciprocal of the base and then changing the sign of the exponent.

Any expression that has negative exponents is not considered to be in simplest form. We will use the definition of a negative exponent and other properties of exponents to write the expression with only positive exponents.

EXAMPLE 4

Simplify: a) {4}^{-2} b) {10}^{-3}.

Solution
a) {4}^{-2}
Use the definition of a negative exponent, {a}^{-n}=\frac{1}{{a}^{n}}. \frac{1}{{4}^{2}}
Simplify. \frac{1}{16}
b) {10}^{-3}
Use the definition of a negative exponent, {a}^{-n}=\frac{1}{{a}^{n}}. \frac{1}{{10}^{3}}
Simplify. \frac{1}{1000}

TRY IT 4

Simplify: a) {2}^{-3} b) {10}^{-7}.

Show answer

a) \frac{1}{8} b) \frac{1}{{10}^{7}}

In (Example 4) we raised an integer to a negative exponent. What happens when we raise a fraction to a negative exponent? We’ll start by looking at what happens to a fraction whose numerator is one and whose denominator is an integer raised to a negative exponent.

\frac{1}{{a}^{\-n}}
Use the definition of a negative exponent, {a}^{-n}=\frac{1}{{a}^{n}}. \frac{1}{\frac{1}{{a}^{n}}}
Simplify the complex fraction. 1\cdot\frac{{a}^{n}}{1}
Multiply. {a}^{n}

This leads to the Property of Negative Exponents.

Property of Negative Exponents

If n is an integer and a\ne 0, then \frac{1}{{a}^{-n}}={a}^{n}.

EXAMPLE 5

Simplify: \frac{1}{{3}^{-2}}.

Solution
\frac{1}{{3}^{-2}}
Use the property of a negative exponent, \frac{1}{{a}^{-n}}={a}^{n}. {3}^{2}
Simplify. 9

TRY IT 5

Simplify: \frac{1}{{4}^{-3}}.

Show answer

 64

Suppose now we have a fraction raised to a negative exponent. Let’s use our definition of negative exponents to lead us to a new property.

{\left(\frac{3}{4}\right)}^{-2}
Use the definition of a negative exponent, {a}^{-n}=\frac{1}{{a}^{n}}. \frac{1}{{\left(\frac{3}{4}\right)}^{2}}
Simplify the denominator. \frac{1}{\frac{9}{16}}
Simplify the complex fraction. \frac{16}{9}
But we know that \frac{16}{9} is {\left(\frac{4}{3}\right)}^{2}.
This tells us that: {\left(\frac{3}{4}\right)}^{-2}={\left(\frac{4}{3}\right)}^{2}

To get from the original fraction raised to a negative exponent to the final result, we took the reciprocal of the base—the fraction—and changed the sign of the exponent.

This leads us to the Quotient to a Negative Power Property.

Quotient to a Negative Exponent Property

If a\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}b are real numbers, a\ne 0,b\ne 0, and n is an integer, then {\left(\frac{a}{b}\right)}^{-n}={\left(\frac{b}{a}\right)}^{n}.

EXAMPLE 6

Simplify:  {\left(\frac{5}{7}\right)}^{-2} .

Solution
{\left(\frac{5}{7}\right)}^{-2}
Use the Quotient to a Negative Exponent Property, {\left(\frac{a}{b}\right)}^{-n}={\left(\frac{b}{a}\right)}^{n}.
Take the reciprocal of the fraction and change the sign of the exponent. {\left(\frac{7}{5}\right)}^{2}
Simplify. \frac{49}{25}

TRY IT 6

Simplify:  {\left(\frac{2}{3}\right)}^{-4} .

Show answer

 \frac{81}{16}

When simplifying an expression with exponents, we must be careful to correctly identify the base.

EXAMPLE 7

Simplify: a) {\left(-3\right)}^{-2} b) -{3}^{-2} c) {\left(-\frac{1}{3}\right)}^{-2} d) -{\left(\frac{1}{3}\right)}^{-2}.

Solution
a) Here the exponent applies to the base -3. {\left(-3\right)}^{-2}
Take the reciprocal of the base and change the sign of the exponent. \frac{1}{{\left(-3\right)}^{-2}}
Simplify. \frac{1}{9}
b) The expression -{3}^{-2} means “find the opposite of {3}^{-2}.” Here the exponent applies to the base {\left(-\frac{1}{3}\right)}^{}. -{3}^{-2}
Rewrite as a product with -1. -1\cdot{3}^{-2}
Take the reciprocal of the base and change the sign of the exponent. -1\cdot\frac{1}{{3}^{2}}
Simplify. -\frac{1}{9}
c) Here the exponent applies to the base {\left(-\frac{1}{3}\right)}^{}. {\left(-\frac{1}{3}\right)}^{-2}
Take the reciprocal of the base and change the sign of the exponent. {\left(-\frac{3}{1}\right)}^{2}
Simplify. 9
d) The expression -{\left(\frac{1}{3}\right)}^{-2} means “find the opposite of {\left(\frac{1}{3}\right)}^{-2}.” Here the exponent applies to the base \left(\frac{1}{3}\right).
Rewrite as a product with -1. -1\cdot{\left(\frac{1}{3}\right)}^{-2}
Take the reciprocal of the base and change the sign of the exponent. -1\cdot{\left(\frac{3}{1}\right)}^{2}
Simplify. -9

TRY IT 7

Simplify: a) {\left(-5\right)}^{-2} b) -{5}^{-2} c) {\left(-\frac{1}{5}\right)}^{-2} d) -{\left(\frac{1}{5}\right)}^{-2}.

Show answer

a) \frac{1}{25} b) -\frac{1}{25} c) 25 d) -25

We must be careful to follow the Order of Operations. In the next example, parts (a) and (b) look similar, but the results are different.

EXAMPLE 8

Simplify: a) 4\cdot{2}^{-1} b) {\left(4\cdot 2\right)}^{-1}.

Solution
a)
Do exponents before multiplication.		 4\cdot{2}^{-1}
Use {a}^{-n}=\frac{1}{{a}^{n}}. 4\cdot\frac{1}{{2}^{1}}
Simplify. 2
b) {\left(4\cdot 2\right)}^{-1}
Simplify inside the parentheses first. {\left(8\right)}^{-1}
Use {a}^{-n}=\frac{1}{{a}^{n}}. \frac{1}{{8}^{1}}
Simplify. \frac{1}{8}

TRY IT 8

Simplify: a) 6\cdot{3}^{-1} b) {\left(6\cdot3\right)}^{-1}.

Show answer

a) 2 b) \frac{1}{18}

Use Formulas with Exponents in Applications

In this section, we will use geometry formulas that contain exponents to solve problems. Since we will be solving applications, we will once again show our Problem-Solving Strategy for Geometry Applications.

Problem Solving Strategy for Geometry Applications

  1. Read the problem and make sure you understand all the words and ideas. Draw the figure and label it with the given information.
  2. Identify what you are looking for.
  3. Name what you are looking for. Choose a variable to represent that quantity.
  4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
  5. Solve the equation using good algebra techniques.
  6. Check the answer in the problem and make sure it makes sense.
  7. Answer the question with a complete sentence.

A cube is a rectangular solid whose length, width, and height are equal. See Volume and Surface Area of a Cube, below. Substituting, s for the length, width and height into the formulas for volume and surface area of a rectangular solid, we get:

\begin{array}{ccccc}V=LWH\hfill & & & & S=2LH+2LW+2WH\hfill \\ V=\text{s}\cdot\text{s}\cdot\text{s}\hfill & & & & S=2\text{s}\cdot\text{s}+2\text{s}\cdot\text{s}+2\text{s}\cdot\text{s}\hfill \\ V={\text{s}}^{3}\hfill & & & & S=2{s}^{2}+2{s}^{2}+2{s}^{2}\hfill \\ & & & & S=6{s}^{2}\hfill \end{array}

So for a cube, the formulas for volume and surface area are V={s}^{3} and S=6{s}^{2}.

Volume and Surface Area of a Cube

For any cube with sides of length s,

An image of a cube is shown. Each side is labeled s. Beside this is Volume: V equals s cubed. Below that is Surface Area: S equals 6 times s squared.

EXAMPLE 3

A cube is 2.5 inches on each side. Find its a) volume and b) surface area.

Solution

Step 1 is the same for both a) and b), so we will show it just once.

Step 1. Read the problem. Draw the figure and
label it with the given information.		 .
a)
Step 2. Identify what you are looking for. the volume of the cube
Step 3. Name. Choose a variable to represent it. let V = volume
Step 4. Translate.
Write the appropriate formula.		 V={s}^{3}
Step 5. Solve. Substitute and solve. V={\left(2.5\right)}^{3}
V=15.625
Step 6. Check: Check your work.
Step 7. Answer the question. The volume is 15.625 cubic inches.
b)
Step 2. Identify what you are looking for. the surface area of the cube
Step 3. Name. Choose a variable to represent it. let S = surface area
Step 4. Translate.
Write the appropriate formula.		 S=6{s}^{2}
Step 5. Solve. Substitute and solve. S=6\cdot {\left(2.5\right)}^{2}
S=37.5
Step 6. Check: The check is left to you.
Step 7. Answer the question. The surface area is 37.5 square inches.

TRY IT 3

For a cube with side 4.5 metres, find the a) volume and b) surface area of the cube.

Show answer
  1. 91.125 cu. m
  2. 121.5 sq. m

EXAMPLE 4

A notepad cube measures 2 inches on each side. Find its a) volume and b) surface area.

Solution
Step 1. Read the problem. Draw the figure and
label it with the given information.		 .
a)
Step 2. Identify what you are looking for. the volume of the cube
Step 3. Name. Choose a variable to represent it. let V = volume
Step 4. Translate.
Write the appropriate formula.		 V={s}^{3}
Step 5. Solve the equation. V={2}^{3}
V=8
Step 6. Check: Check that you did the calculations
correctly.
Step 7. Answer the question. The volume is 8 cubic inches.
b)
Step 2. Identify what you are looking for. the surface area of the cube
Step 3. Name. Choose a variable to represent it. let S = surface area
Step 4. Translate.
Write the appropriate formula.		 S=6{s}^{2}
Step 5. Solve the equation. S=6\cdot {2}^{2}
S=24
Step 6. Check: The check is left to you.
Step 7. Answer the question. The surface area is 24 square inches.

TRY IT 4

A packing box is a cube measuring 4 feet on each side. Find its a) volume and b) surface area.

Show answer
  1. 64 cu. ft
  2. 96 sq. ft

A sphere is the shape of a basketball, like a three-dimensional circle. Just like a circle, the size of a sphere is determined by its radius, which is the distance from the center of the sphere to any point on its surface. The formulas for the volume and surface area of a sphere are given below.

Showing where these formulas come from, like we did for a rectangular solid, is beyond the scope of this course. We will approximate \pi with 3.14.

Volume and Surface Area of a Sphere

For a sphere with radius r\text{:}

An image of a sphere is shown. The radius is labeled r. Beside this is Volume: V equals four-thirds times pi times r cubed. Below that is Surface Area: S equals 4 times pi times r squared.

EXAMPLE 5

A sphere has a radius 6 inches. Find its a) volume and b) surface area.

Solution

Step 1 is the same for both a) and b), so we will show it just once.

Step 1. Read the problem. Draw the figure and label
it with the given information.		 .
a)
Step 2. Identify what you are looking for. the volume of the sphere
Step 3. Name. Choose a variable to represent it. let V = volume
Step 4. Translate.
Write the appropriate formula.		 V=\frac{4}{3}\pi {r}^{3}
Step 5. Solve. V\approx \frac{4}{3}\left(3.14\right){6}^{3}
V\approx 904.32\phantom{\rule{0.2em}{0ex}}\text{cubic inches}
Step 6. Check: Double-check your math on a calculator.
Step 7. Answer the question. The volume is approximately 904.32 cubic inches.
b)
Step 2. Identify what you are looking for. the surface area of the cube
Step 3. Name. Choose a variable to represent it. let S = surface area
Step 4. Translate.
Write the appropriate formula.		 S=4\pi {r}^{2}
Step 5. Solve. S\approx 4\left(3.14\right){6}^{2}
S\approx 452.16
Step 6. Check: Double-check your math on a calculator
Step 7. Answer the question. The surface area is approximately 452.16 square inches.

TRY IT 5

Find the a) volume and b) surface area of a sphere with radius 3 centimetres.

Show answer
  1. 113.04 cu. cm
  2. 113.04 sq. cm

EXAMPLE 6

A globe of Earth is in the shape of a sphere with radius 14 centimetres. Find its a) volume and b) surface area. Round the answer to the nearest hundredth.

Solution
Step 1. Read the problem. Draw a figure with the
given information and label it.		 .
a)
Step 2. Identify what you are looking for. the volume of the sphere
Step 3. Name. Choose a variable to represent it. let V = volume
Step 4. Translate.
Write the appropriate formula.
Substitute. (Use 3.14 for \pi)		 V=\frac{4}{3}\pi {r}^{3}
V\approx \frac{4}{3}\left(3.14\right){14}^{3}
Step 5. Solve. V\approx 11,488.21
Step 6. Check: We leave it to you to check your calculations.
Step 7. Answer the question. The volume is approximately 11,488.21 cubic inches.
b)
Step 2. Identify what you are looking for. the surface area of the sphere
Step 3. Name. Choose a variable to represent it. let S = surface area
Step 4. Translate.
Write the appropriate formula.
Substitute. (Use 3.14 for \pi)		 S=4\pi {r}^{2}
S\approx 4\left(3.14\right){14}^{2}
Step 5. Solve. S\approx 2461.76
Step 6. Check: We leave it to you to check your calculations.
Step 7. Answer the question. The surface area is approximately 2461.76 square inches.

TRY IT 6

A beach ball is in the shape of a sphere with radius of 9 inches. Find its a) volume and b) surface area.

Show answer
  1. 3052.08 cu. in.
  2. 1017.36 sq. in.

Convert from Decimal Notation to Scientific Notation

Remember working with place value for whole numbers and decimals? Our number system is based on powers of 10. We use tens, hundreds, thousands, and so on. Our decimal numbers are also based on powers of tens—tenths, hundredths, thousandths, and so on. Consider the numbers 4,000 and 0.004. We know that 4,000 means 4\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}1,000 and 0.004 means 4\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}\frac{1}{1,000}.

If we write the 1000 as a power of ten in exponential form, we can rewrite these numbers in this way:

\begin{array}{cccc}4,000\hfill & & & \phantom{\rule{4em}{0ex}}0.004\hfill \\ 4\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}1,000\hfill & & & \phantom{\rule{4em}{0ex}}4\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}\frac{1}{1,000}\hfill \\ 4\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{3}\hfill & & & \phantom{\rule{4em}{0ex}}4\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}\frac{1}{{10}^{3}}\hfill \\ & & & \phantom{\rule{4em}{0ex}}4\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{-3}\hfill \end{array}

When a number is written as a product of two numbers, where the first factor is a number greater than or equal to one but less than 10, and the second factor is a power of 10 written in exponential form, it is said to be in scientific notation.

Scientific Notation

A number is expressed in scientific notation when it is of the form

\begin{array}{cccc}& & & a\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{n}\phantom{\rule{0.2em}{0ex}}\text{where}\phantom{\rule{0.2em}{0ex}}1\le a<10\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}n\phantom{\rule{0.2em}{0ex}}\text{is an integer}\hfill \end{array}

It is customary in scientific notation to use as the \phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}} multiplication sign, even though we avoid using this sign elsewhere in algebra.

If we look at what happened to the decimal point, we can see a method to easily convert from decimal notation to scientific notation.

This figure illustrates how to convert a number to scientific notation. It has two columns. In the first column is 4000 equals 4 times 10 to the third power. Below this, the equation is repeated, with an arrow demonstrating that the decimal point at the end of 4000 has moved three places to the left, so that 4000 becomes 4.000. The second column has 0.004 equals 4 times 10 to the negative third power. Below this, the equation is repeated, with an arrow demonstrating how the decimal point in 0.004 is moved three places to the right to produce 4.

In both cases, the decimal was moved 3 places to get the first factor between 1 and 10

\begin{array}{cccc}\text{The power of 10 is positive when the number is larger than 1:}\hfill & & & 4,000=4\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{3}\hfill \\ \text{The power of 10 is negative when the number is between 0 and 1:}\hfill & & & 0.004=4\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{-3}\hfill \end{array}

EXAMPLE 9

How to Convert from Decimal Notation to Scientific Notation

Write in scientific notation: 37,000.

Solution

This figure is a table that has three columns and four rows. The first column is a header column, and it contains the names and numbers of each step. The second column contains further written instructions. The third column contains math. On the top row of the table, the first cell on the left reads “Step 1. Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.” The second cell reads “Remember, there is a decimal at the end of 37,000.” The third cell contains 37,000. One line down, the second cell reads “Move the decimal after the 3. 3.7000 is between 1 and 10.”In the second row, the first cell reads “Step 2. Count the number of decimal places, n, that the decimal place was moved. The second cell reads “The decimal point was moved 4 places to the left.” The third cell contains 370000 again, with an arrow showing the decimal point jumping places to the left from the end of the number until it ends up between the 3 and the 7.In the third row, the first cell reads “Step 3. Write the number as a product with a power of 10. If the original number is greater than 1, the power of 10 will be 10 to the n power. If it’s between 0 and 1, the power of 10 will be 10 to the negative n power.” The second cell reads “37,000 is greater than 1, so the power of 10 will have exponent 4.” The third cell contains 3.7 times 10 to the fourth power.In the fourth row, the first cell reads “Step 4. Check.” The second cell reads “Check to see if your answer makes sense.” The third cell reads “10 to the fourth power is 10,000 and 10,000 times 3.7 will be 37,000.” Below this is 37,000 equals 3.7 times 10 to the fourth power.

TRY IT 9

Write in scientific notation: 96,000.

Show answer

9.6\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{4}

HOW TO: Convert from decimal notation to scientific notation
  1. Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.
  2. Count the number of decimal places, n, that the decimal point was moved.
  3. Write the number as a product with a power of 10.
    If the original number is:
  • greater than 1, the power of 10 will be 10n.
  • between 0 and 1, the power of 10 will be 10−n.
  • Check.

EXAMPLE 10

Write in scientific notation: 0.0052.

Solution

The original number, 0.0052, is between 0 and 1 so we will have a negative power of 10

0.0052.
Move the decimal point to get 5.2, a number between 1 and 10. 0.0052, with an arrow showing the decimal point jumping three places to the right until it ends up between the 5 and 2.
Count the number of decimal places the point was moved. 3 places.
Write as a product with a power of 10. 5.2 times 10 to the power of negative 3.
Check.
\begin{array}{ccccc}\\ \\ \phantom{\rule{3em}{0ex}}5.2\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{-3}\hfill \\ \phantom{\rule{3em}{0ex}}5.2\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}\frac{1}{{10}^{3}}\hfill \\ \phantom{\rule{3em}{0ex}}5.2\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}\frac{1}{1000}\hfill \\ \phantom{\rule{3em}{0ex}}5.2\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}0.001\hfill \end{array}
\phantom{\rule{2em}{0ex}}0.0052

TRY IT 10

Write in scientific notation: 0.0078.

Show answer

7.8\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{-3}

Convert Scientific Notation to Decimal Form

How can we convert from scientific notation to decimal form? Let’s look at two numbers written in scientific notation and see.

\begin{array}{cccc}\hfill 9.12\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{4}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}9.12\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{-4}\hfill \\ \hfill 9.12\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}10,000\hfill & & & \hfill \phantom{\rule{4em}{0ex}}9.12\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}0.0001\hfill \\ \hfill 91,200\hfill & & & \hfill \phantom{\rule{4em}{0ex}}0.000912\hfill \end{array}

If we look at the location of the decimal point, we can see an easy method to convert a number from scientific notation to decimal form.

\begin{array}{cccc}\hfill 9.12\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{4}=91,200\hfill & & & \hfill \phantom{\rule{4em}{0ex}}9.12\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{-4}=0.000912\hfill \end{array}

This figure has two columns. In the left column is 9.12 times 10 to the fourth power equals 91,200. Below this, the same scientific notation is repeated, with an arrow showing the decimal point in 9.12 being moved four places to the right. Because there are no digits after 2, the final two places are represented by blank spaces. Below this is the text “Move the decimal point four places to the right.” In the right column is 9.12 times 10 to the negative fourth power equals 0.000912. Below this, the same scientific notation is repeated, with an arrow showing the decimal point in 9.12 being moved four places to the left. Because there are no digits before 9, the remaining three places are represented by spaces. Below this is the text “Move the decimal point 4 places to the left.”

In both cases the decimal point moved 4 places. When the exponent was positive, the decimal moved to the right. When the exponent was negative, the decimal point moved to the left.

EXAMPLE 11

How to Convert Scientific Notation to Decimal Form

Convert to decimal form: 6.2\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{3}.

Solution

This figure is a table that has three columns and three rows. The first column is a header column, and it contains the names and numbers of each step. The second column contains further written instructions. The third column contains math. On the top row of the table, the first cell on the left reads “Step 1. Determine the exponent, n, on the factor 10.” The second cell reads “The exponent is 3.” The third cell contains 6.2 times 10 cubed.In the second row, the first cell reads “Step 2. Move the decimal n places, adding zeros if needed. If the exponent is positive, move the decimal point n places to the right. If the exponent is negative, move the decimal point absolute value of n places to the left.” The second cell reads “The exponent is positive so move the decimal point 3 places to the right. We need to add two zeros as placeholders.” The third cell contains 6.200, with an arrow showing the decimal point jumping places to the right, from between the 6 and 2 to after the second 00 in 6.200. Below this is the number 6,200.In the third row, the first cell reads “Step 3. Check to see if your answer makes sense.” The second cell is blank. The third reads “10 cubed is 1000 and 1000 times 6.2 will be 6,200.” Beneath this is 6.2 times 10 cubed equals 6,200.

TRY IT 11

Convert to decimal form: 1.3\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{3}.

Show answer

1,300

The steps are summarized below.

HOW TO: Convert scientific notation to decimal form.

To convert scientific notation to decimal form:

  1. Determine the exponent, n, on the factor 10.
  2. Move the decimal n places, adding zeros if needed.
    • If the exponent is positive, move the decimal point n places to the right.
    • If the exponent is negative, move the decimal point |n| places to the left.
  3. Check.

EXAMPLE 12

Convert to decimal form: 8.9\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{-2}.

Solution
8.9 times 10 to the power of negative 2.
Determine the exponent, n, on the factor 10. The exponent is negative 2.
Since the exponent is negative, move the decimal point 2 places to the left. 8.9, with an arrow the decimal place showing the decimal point being moved two places to the left.
Add zeros as needed for placeholders. 8.9 times 10 to the power of negative 2 equals 0.089.

TRY IT 12

Convert to decimal form: 1.2\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{-4}.

Show answer

0.00012

Multiply and Divide Using Scientific Notation

Astronomers use very large numbers to describe distances in the universe and ages of stars and planets. Chemists use very small numbers to describe the size of an atom or the charge on an electron. When scientists perform calculations with very large or very small numbers, they use scientific notation. Scientific notation provides a way for the calculations to be done without writing a lot of zeros. We will see how the Properties of Exponents are used to multiply and divide numbers in scientific notation.

EXAMPLE 13

Multiply. Write answers in decimal form: \left(4\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{5}\right)\left(2\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{-7}\right).

Solution
\left(4\times{10}^{5}\right)\left(2\times{10}^{-7}\right)
Use the Commutative Property to rearrange the factors. 4\cdot 2\cdot{10}^{5}\cdot{10}^{-7}
Multiply. 8\times{10}^{-2}
Change to decimal form by moving the decimal two places left. 0.08

TRY IT 13

Multiply \left(3\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{6}\right)\left(2\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{-8}\right). Write answers in decimal form.

Show answer

0.06

EXAMPLE 14

Multiply. Write answer in scientific notation: \left(3.2\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{-3}\right)\left(6.3\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{9}\right).

Solution
\left(3.2\times{10}^{-3}\right)\left(6.3\times{10}^{9}\right)
Use the Commutative Property to rearrange the factors. 3.2\cdot 6.3\cdot{10}^{-3}\cdot{10}^{9}
Multiply. 20.16\times{10}^{6}
Write the answer in scientific notation. 2.016\times{10}^{7}

 

TRY IT 14

Multiply. Write answer in scientific notation: \left(2.7\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{-6}\right)\left(7.9\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{3}\right).

Show answer

\(2.133\times{10}^{-2}\)

EXAMPLE 15

Divide. Write answers in decimal form: \frac{9\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{3}}{3\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{-2}}.

Solution
\frac{9\times{10}^{3}}{3\times{10}^{-2}}
Separate the factors, rewriting as the product of two fractions. \frac{9}{3}\times\frac{{10}^{3}}{{10}^{-2}}
Divide.  3\times{10}^{5}
Change to decimal form by moving the decimal five places right.  300,000

TRY IT 15

Divide \frac{8\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{4}}{2\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{-1}}. Write answers in decimal form.

Show answer

400,000

EXAMPLE 16

Divide. Write answer in scientific notation: \frac{3.2\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{4}}{8\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{-5}}.

Solution
\frac{3.2\times{10}^{4}}{8times{10}^{-5}}
Separate the factors, rewriting as the product of two fractions. \frac{3.2}{8}\times\frac{{10}^{4}}{{10}^{-5}}
Divide.  0.4\times{10}^{3}
Write answer in scientific notation  4\times{10}^2

TRY IT 16

Divide \frac{2.585\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{5}}{3.8\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{-1}}. Write answer in scientific notation.

Show answer

6.8\times{10}^3

Access these online resources for additional instruction and practice with integer exponents and scientific notation:

Key Concepts

  • Exponential Notation
    This figure has two columns. In the left column is a to the m power. The m is labeled in blue as an exponent. The a is labeled in red as the base. In the right column is the text “a to the m powder means multiply m factors of a.” Below this is a to the m power equals a times a times a times a, followed by an ellipsis, with “m factors” written below in blue.
  • Product Property of Exponents
    • If a,b are real numbers and m,n are whole numbers, then
      \ {a}^{m}\cdot {a}^{n}& = {a}^{m+n}
  • Quotient Property for Exponents:
    • If a is a real number, a\ne 0, and m,n are whole numbers, then:
      \dfrac{{a}^{m}}{{a}^{n}}={a}^{m-n},m > n\text{ and }\dfrac{{a}^{m}}{{a}^{n}}=\dfrac{1}{{a}^{m-n}},n > m
  • Zero Exponent
    • If a is a non-zero number, then {a}^{0}=1.
  • Property of Negative Exponents
    • If n is a positive integer and a\ne 0, then \frac{1}{{a}^{-n}}={a}^{n}
  • Quotient to a Negative Exponent
    • If a,b are real numbers, b\ne 0 and n is an integer , then {\left(\frac{a}{b}\right)}^{-n}={\left(\frac{b}{a}\right)}^{n}
  • To convert a decimal to scientific notation:
    1. Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.
    2. Count the number of decimal places, n, that the decimal point was moved.
    3. Write the number as a product with a power of 10. If the original number is:
      • greater than 1, the power of 10 will be {10}^{n}
      • between 0 and 1, the power of 10 will be {10}^{-n}
    4. Check.
  • To convert scientific notation to decimal form:
    1. Determine the exponent, n, on the factor 10.
    2. Move the decimal nplaces, adding zeros if needed.
      • If the exponent is positive, move the decimal point n places to the right.
      • If the exponent is negative, move the decimal point |n| places to the left.
    3. Check

1.5 Exercise Set

In the following exercises, simplify each expression with exponents.

    1. {3}^{5}
    2. {9}^{1}
    3. {\left(\dfrac{1}{3}\right)}^{2}
    4. {\left(0.2\right)}^{4}
    1. {2}^{6}
    2. {14}^{1}
    3. {\left(\dfrac{2}{5}\right)}^{3}
    4. {\left(0.7\right)}^{2}
    1. {\left(-6\right)}^{4}
    2. -{6}^{4}
    1. -{\left(\dfrac{2}{3}\right)}^{2}
    2. {\left(-\dfrac{2}{3}\right)}^{2}
    1. -{0.5}^{2}
    2. {\left(-0.5\right)}^{2}
In the following exercises, simplify.
    1. {20}^{0}
    2. {b}^{0}
    1. {\left(-27\right)}^{0}
    2. -\left({27}^{0}\right)

In the following exercises, simplify.

    1. {3}^{-4}
    2. {10}^{-2}
    1. {2}^{-8}
    2. {10}^{-2}
    1. \frac{1}{{5}^{-2}}
    2.  \frac{1}{{10}^{-4}}
    3.  {\left(\frac{3}{10}\right)}^{-2}
    4. {\left(\frac{7}{2}\right)}^{-3}
    1. {\left(-7\right)}^{-2}
    2. -{7}^{-2}
    3. {\left(-\frac{1}{7}\right)}^{-2}
    4. -{\left(\frac{1}{7}\right)}^{-2}
    1. -{5}^{-3}
    2. {\left(-\frac{1}{5}\right)}^{-3}
    3. -{\left(\frac{1}{5}\right)}^{-3}
    4. {\left(-5\right)}^{-3}
    1. 2\cdot{5}^{-1}
    2. {\left(2\cdot 5\right)}^{-1}
    1. 3\cdot{4}^{-2}
    2. {\left(3\cdot 4\right)}^{-2}

In the following exercises, find a) the volume and b) the surface area of the cube with the given side length.

  1. 5 centimetres
  2. 10.4 feet

In the following exercises, solve.

  1. Museum A cube-shaped museum has sides 64 metres long. Find its a) volume and b) surface area.
  2. Base of statue The base of a statue is a cube with sides 2.8 metres long. Find its a) volume and b) surface area.

In the following exercises, find a) the volume and b) the surface area of the sphere with the given radius. Round answers to the nearest hundredth.

  1. 3 centimetres
  2. 7.5 feet

In the following exercises, solve. Round answers to the nearest hundredth.

  1. Exercise ball An exercise ball has a radius of 15 inches. Find its a) volume and b) surface area.
  2. Golf ball A golf ball has a radius of 4.5 centimetres. Find its a) volume and b) surface area.

In the following exercises, write each number in scientific notation.

  1. 340,000
  2. 1,290,000
  3. 0.041
  4. 0.00000103

In the following exercises, convert each number to decimal form.

  1. 8.3\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{2}
  2. 1.6\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{10}
  3. 3.8\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{-2}
  4. 1.93\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{-5}

In the following exercises, multiply. Write your answer in decimal form.

  1. \left(2\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{2}\right)\left(1\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{-4}\right)
  2. \left(3.5\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{-4}\right)\left(1.6\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{-2}\right)

In the following exercises, divide. Write your answer in decimal form.

  1. \frac{5\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{-2}}{1\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{-10}}
  2. \frac{8\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{6}}{4\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{-1}}
  3. The population of the world on July 1, 2010 was more than 6,850,000,000. Write the number in scientific notation
  4. The probability of winning the lottery was about 0.0000000057. Write the number in scientific notation.
  5. The width of a proton is 1\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{-5} of the width of an atom. Convert this number to decimal form.
  6. Coin production In 1942, the U.S. Mint produced 154,500,000 nickels. Write 154,500,000 in scientific notation.

  7. Debt At the end of fiscal year 2019 the gross Canadian federal government debt was estimated to be approximately $685,450,000,000 ($685.45 billion), according to the Federal Budget. The population of Canada was approximately 37,590,000 people at the end of fiscal year 2019

    a) Write the debt in scientific notation.

    b) Write the population in scientific notation.

    c) Find the amount of debt per person by using scientific notation to divide the debt by the population. Write the answer in scientific notation.

Answers:

    1. 243
    2. 9
    3. \frac{1}{9}
    4. 0.0016
    1. 64
    2. 14
    3. \frac{8}{125}
    4. 0.49
    1. 1296
    2. -1296
    1. -\frac{4}{9}
    2. \frac{4}{9}
    1. -0.25
    2. 0.25
    1. 1
    2. 1
    1. 1
    2. -1
    1. \frac{1}{81}
    2. \frac{1}{100}
    1. \frac{1}{256}
    2. \frac{1}{100}
  10. 25
  11. 10000
  12. \frac{100}{9}
  13. \frac{8}{343}
    1. \frac{1}{49}
    2. -\frac{1}{49}
    3. 49
    4. -49
    1. -\frac{1}{125}
    2. -125
    3. -125
    4. -\frac{1}{125}
    1. \frac{2}{5}
    2. \frac{1}{10}
    1. \frac{3}{16}
    2. \frac{1}{144}
    1. 125 cu. cm
    2. 150 sq. cm
    1. 1124.864 cu. ft.
    2. 648.96 sq. ft
    1. 262,144 cu. ft
    2. 24,576 sq. ft
    1. 21.952 cu. m
    2. 47.04 sq. m
    1. 113.04 cu. cm
    2. 113.04 sq. cm
    1. 1,766.25 cu. ft
    2. 706.5 sq. ft
    1. 14,130 cu. in.
    2. 2,826 sq. in.
    1. 381.51 cu. cm
    2. 254.34 sq. cm
  26. 3.4\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{5}
  27. 1.29\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{6}
  28. 4.1\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{-2}
  29. 1.03\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{-6}
  30. 830
  31. 16,000,000,000
  32. 0.038
  33. 0.0000193
  34. 0.02
  35. 5.6\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{-6}
  36. 500,000,000
  37. 20,000,000
  38. 6.85\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{9}.
  39. 5.7\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{-10}
  40. 0.00001
  41. 1.545\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{8}
    1. 1.86\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{13}
    2. 3\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{8}
    3. 6.2\phantom{\rule{0.2em}{0ex}}\times\phantom{\rule{0.2em}{0ex}}{10}^{4}

Attributions

This chapter has been adapted from “Integer Exponents and Scientific Notation” in Elementary Algebra (OpenStax) by Lynn Marecek and MaryAnne Anthony-Smith, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

License

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Business/Technical Mathematics Copyright © 2021 by Izabela Mazur and Kim Moshenko is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.

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