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Section 9.2 Negative Exponents and Scientific Notation

Subsection Zero as an Exponent

Negative exponents simplify many calculations and give us a useful form for writing very large or very small numbers.

But first, what do negative exponents mean? In order for them to be useful, their properties must fit with what we already know about exponents.

We know that \(a^n\) means the product of \(n\) factors of \(a\text{.}\) For example, \(a^3\) means \(a \cdot a \cdot a\text{.}\) Is there a reasonable meaning for the power \(a^0\text{?}\)

Consider the quotient \(\dfrac{a^4}{a^4}\text{.}\) If \(a\) is not zero, this quotient equals 1, because any nonzero number divided by itself is 1. On the other hand, we can also think of \(\dfrac{a^4}{a^4}\) as a quotient of powers and subtract the exponents. If we extend the second law of exponents to include the case \(m=n\text{,}\) we have

\begin{equation*} \dfrac{a^4}{a^4} = a^{4-4} = a^0 \end{equation*}

Thus, it seems reasonable to make the following definition.

Zero as an Exponent.
\begin{equation*} \blert{a^0=1,~~~~\text{if}~~~~a \not= 0} \end{equation*}

For example,

\begin{equation*} 3^0=1,~~~~(-427)^0=1,~~~~\text{and} ~~~~(5xy)^0=1~~~\text{if}~~x,y \not= 0 \end{equation*}

Reading Questions Reading Questions

1.

Why does it make sense that \(a^0=1\) (as long as \(a \not= 0\))?

Subsection Negative Exponents

We can also use the second law of exponents to give meaning to negative exponents. Consider the quotient \(\dfrac{a^4}{a^7}\text{.}\) According to the second law of exponents,

\begin{equation*} \dfrac{a^4}{a^7} = \dfrac{1}{a^{7-4}} = \dfrac{1}{a^3} \end{equation*}

However, if we allow negative numbers as exponents, we can apply the first half of the second law to obtain

\begin{equation*} \dfrac{a^4}{a^7} = a^{4-7} = a^{-3} \end{equation*}

We therefore define \(a^{-3}\) to mean \(\dfrac{1}{a^3}\text{.}\) In general, for \(a \not= 0\text{,}\) we make the following definition.

Negative Exponents.
\begin{equation*} \blert{a^{-n} = \dfrac{1}{a^n}~~~~\text{if}~~~~a \not= 0} \end{equation*}

For example,

\begin{equation*} 2^{-4} = \dfrac{1}{2^4} ~~~~~~\text{and}~~~~~~x^{-5} = \dfrac{1}{x^5} \end{equation*}

A negative exponent indicates the reciprocal of the power with the positive exponent.

Example 9.10.

Write each expression without exponents.

  1. \(10^{-4}\)
  2. \((\dfrac{1}{4})^{-3}\)
  3. \((\dfrac{3}{5})^{-2}\)
Solution
  1. \(10^{-4} = \dfrac{1}{10^4} = \dfrac{1}{10,000},\) or 0.0001
  2. To compute a negative power of a fraction, we compute the corresponding positive power of its reciprocal. (This is because of the fifth law of exponents.)

    \begin{equation*} (\dfrac{1}{4})^{-3} = 4^3 = 64 \end{equation*}
  3. As in part (b), we compute the corresponding positive power of the reciprocal of \(\dfrac{3}{5}\text{.}\)

    \begin{equation*} (\dfrac{3}{5})^{-2}=(\dfrac{5}{3})^2 = \dfrac{25}{9} \end{equation*}

Reading Questions Reading Questions

2.

What does a negative exponent mean?

Caution 9.11.

A negative exponent does not mean that the power is negative. For example,

\begin{equation*} 2^{-4} \not= -2^4 \end{equation*}

We can also rewrite fractions using negative exponents.

Example 9.12.

Write each expression using negative exponents.

  1. \(\dfrac{1}{3^4}\)
  2. \(\dfrac{7}{10^2}\)
  3. \(\dfrac{8}{x}\)
Solution
  1. \(\dfrac{1}{3^4} = 3^{-4}\)
  2. \(\dfrac{7}{10^2} = 7 \cdot \dfrac{1}{10^2} = 7 \cdot 10^{-2}\)
  3. \(\dfrac{8}{x} = 8 \cdot \dfrac{1}{x} = 8x^{-1}\)
Caution 9.13.

An exponent applies only to its base. For example, the exponent \(-2\) in the expression \(3x^{-2}\) applies only to \(x\text{,}\) but in \((3x)^{-2}\text{,}\) the exponent applies to \(3x\text{.}\) Thus,

\begin{equation*} 3x^{-2} = 3 \cdot \dfrac{1}{x^2} = \dfrac{3}{x^2}~~~~~~\text{but}~~~~~~(3x)^{-2} = \dfrac{1}{(3x)^2} = \dfrac{1}{9x^2} \end{equation*}

Subsection Laws of Exponents

The laws of exponents apply to negative exponents. In particular, if we allow negative exponents, we can write the second law as a single rule.

Laws of Exponents.
  1. \begin{equation*} \blert{a^m \cdot a^n = a^{m+n}} \end{equation*}
  2. \begin{equation*} \blert{\dfrac{a^m}{a^n} = a^{m-n}~~~~(a \not= 0)} \end{equation*}
  3. \begin{equation*} \blert{\left(a^m\right)^n = a^{mn}} \end{equation*}
  4. \begin{equation*} \blert{(ab)^n = a^nb^n} \end{equation*}
  5. \begin{equation*} \blert{\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}} \end{equation*}
Example 9.14.

Simplify by using the first or second law of exponents.

  1. \(x^5 \cdot x^{-8}\)
  2. \(\dfrac{5^2}{5^{-6}}\)
Solution
  1. Apply the first law: add the exponents.

    \begin{equation*} x^5 \cdot x^{-8} = x^{5-8} = x^{-3} \end{equation*}
  2. Apply the second law: subtract the exponents.

    \begin{equation*} \dfrac{5^2}{5^{-6}} = 5^{2-(-6)} = 5^8 \end{equation*}
Look Closer.

As a special case of the second law, note that \(1=a^0\text{,}\) so we can write

\begin{equation*} \dfrac{1}{a^{-n}} = \dfrac{a^0}{a^{-n}} = a^{0-(-n)} = a^n \end{equation*}

Thus,

\begin{equation*} \blert{\dfrac{1}{a^{-n}} = a^n}~~~~\text{and}~~~~\blert{\dfrac{b}{a^{-n}} = ba^n,~~a \not= 0} \end{equation*}
Example 9.15.
  1. \(\dfrac{1}{2^{-3}} = 2^3\)
  2. \(\dfrac{8}{x^{-5}} = 8x^5\)

Here are some examples using the laws of exponents.

Example 9.16.

Simplify by using the third or fourth law of exponents.

  1. \(\left(2^{-3}\right)^{-3}\)
  2. \((ab)^{-3}\)
Solution
  1. Apply the third law: multiply the exponents.

    \begin{equation*} \left(2^{-3}\right)^{-3} = x^{5-8} = 2^{-3(-3)}=2^9 \end{equation*}
  2. Apply the fourth law: raise each factor to the power.

    \begin{equation*} (ab)^{-3} = a^{-3}b^{-3} \end{equation*}

Reading Questions Reading Questions

3.

What is the difference between \(2^{-3}\) and \(-2^3\text{?}\)

Subsection Powers of Ten

Scientists and engineers often encounter very large numbers such as

\begin{equation*} 5,980,000,000,000,000,000,000,000 \end{equation*}

(the mass of the earth in kilograms) and very small numbers such as

\begin{equation*} 0.000~ 000~ 000~ 000~ 000~ 000~ 000~ 00167 \end{equation*}

(the mass of a hydrogen atom in grams) in their work. These numbers can be written in a more compact and useful form using powers of 10.

Recall the following facts about our base-10 number system.

Multiplying by Powers of Ten.
  • Multiplying a number by a positive power of 10 has the effect of moving the decimal point \(k\) places to the right, where \(k\) is the exponent on 10. For example,

    \begin{equation*} 2.358 \times 10^2 = 235.8~~~~~~\text{and}~~~~~~17 \times 10^4 = 170,000 \end{equation*}
  • Multiplying a number by a negative power of 10 has the effect of moving the decimal point to the left. For example,

    \begin{equation*} 5452 \times 10^{-3} = 5.452~~~~~~\text{and}~~~~~~2.3 \times 10^{-5} = 0.000~ 023 \end{equation*}

Reading Questions Reading Questions

4.

What happens to a number when we multiply it by a power of 10?

We can also reverse the process above to write a number in factored form, where one factor is a power of 10.

Example 9.17.

Fill in the correct power of 10 for each factored form.

  1. \(38,400 = 3.84 \times \underline{\hspace{2.727272727272727em}}\)
  2. \(0.0057 = 5.7 \times \underline{\hspace{2.727272727272727em}}\)
Solution
  1. To recover \(38,400\) from \(3.84\text{,}\) we must move the decimal point four places to the right, so we multiply by \(10^4\text{.}\) Thus,

    \begin{equation*} 38,400 = 3.84 \times 10^4 \end{equation*}
  2. To recover \(0.0057\) from \(5.7\text{,}\) we must move the decimal point three places to the left, so we multiply by \(10^{-3}\text{.}\) Thus,

    \begin{equation*} 0.0057 = 5.7 \times 10^{-3} \end{equation*}

Subsection Scientific Notation

If there is just one digit to the left of the decimal point in the factored form, we say that the number is written in scientific notation.

A number is in scientific notation if it is expressed as the product of a number between 1 and 10 and a power of 10.

For example,

\begin{equation*} 4.18 \times 10^{12},~~~~ 2.9 \times 10^{-8},~~~~\text{and}~~~~ 4 \times 10^1 \end{equation*}

are written in scientific notation.

To write a number in scientific notation, we first position the decimal point and then determine the correct power of 10.

To Write a Number in Scientific Notation.
  1. Locate the decimal point so that there is exactly one nonzero digit to its left.
  2. Count the number of places you moved the decimal point: this determines the power of 10.

    1. If the original number is greater than 10, the exponent is positive.
    2. If the original number is less than 1, the exponent is negative.
Example 9.18.

Write each number in scientific notation.

  1. \(62,000,000\)
  2. \(0.000431\)
Solution
  1. We position the decimal point so that there is just one nonzero digit to the left of the decimal.

    \begin{equation*} 62,000,000 = 6.2 \times \underline{\hspace{2.727272727272727em}} \end{equation*}

    To recover 62,000,000 from 6.2, we move the decimal point seven places to the right. Therefore, we multiply 6.2 by \(10^7\text{.}\)

    \begin{equation*} 62,000,000 = 6.2 \times 10^7 \end{equation*}
  2. We position the decimal point so that there is just one nonzero digit to the left of the decimal.

    \begin{equation*} 0.000431 = 4.31 \times \underline{\hspace{2.727272727272727em}} \end{equation*}

    To recover 0.000431 from 4.31, we move the decimal point four places to the left. Therefore, we multiply 4.31 by \(10^{-4}\text{.}\)

    \begin{equation*} 0.000431 = 4.31 \times 10^{-4} \end{equation*}
Look Closer.

The easiest way to remember which way to move the decimal point is to note whether the number is large (greater than 10) or small (less than 1). For example,

  • 62,000,000 is a large number, so the exponent on 10 in its scientific form is positive. On the other hand,
  • 0.000431 is a decimal fraction less than 1, so the exponent on 10 in its scientific form is negative.

Reading Questions Reading Questions

5.

How can you tell whether the exponent in scientific notation should be positive or negative?

Subsection Calculators and Scientific Notation

Your calculator displays numbers in scientific notation if they have too many digits to fit in the display screen.

Example 9.19.

Use a calculator to compute the square of 12,345,678.

Solution

We enter

\begin{equation*} 12345678~~\boxed{~x^2~} \end{equation*}

(and press enter on a graphing calculator.) A scientific calculator displays the result as

\begin{equation*} \boxed{1.524157653~~~~14} \end{equation*}

and a graphing calculator displays

\begin{equation*} 1.524157653~~\text{E}~~14 \end{equation*}

(Some calculators may round the result to fewer decimal places.) Both of these displays represent the number \(1.524157653 \times 10^{14}\text{.}\)

Look Closer.

Because scientific notation always involves a power of 10, most calculators do not display the 10, but only its exponent. If you now press the \(\boxed{~\dfrac{1}{x}~}\) key, your calculator will display the reciprocal of \(1.524157653 \times 10^{14}\) as

\begin{equation*} \boxed{6.56100108~~~~-15}~~~~~~\text{or}~~~~~~ 6.56100108~~\text{E}~~-15 \end{equation*}

This is how your calculator displays the number \(6.56100108 \times 10^{-15}\text{.}\)

Caution 9.20.

Do not use your calculator's notation when giving an answer in scientific notation. For example, if your calculator reports a result as 3.47 E 8, you should write \(3.47 \times 10^8\text{.}\)

Reading Questions Reading Questions

6.

How does your calculator display scientific notation?

Your calculator has a special key for entering numbers in scientific notation.

To enter a number in scientific notation into the calculator, we use the key marked either \(\boxed{\text{EXP}}\) or \(\boxed{\text{EE}}\text{.}\)

For instance, to enter \(6.02 \times 10^{23}\text{,}\) we key in the sequence

\begin{equation*} 6.02~~\boxed{\text{EXP}}~~23 \end{equation*}

To enter a number with a negative exponent, such as \(1.66 \times 10^{-27}\text{,}\) on a scientific calculator we key in

\begin{equation*} 1.66~~ \boxed{\text{EXP}}~~ 27~~ \boxed{~\pm~} \end{equation*}

On a graphing calculator we key in

\begin{equation*} 1.66~~\boxed{\text{EE}}~~\boxed{~(-)~}~~ 27 \end{equation*}
Example 9.21.

The People's Republic of China encompasses about 2,317,400,000 acres of land and has a population of 1,335,300,000 people. How many acres of land per person are there in China?

Solution

We divide the number of acres of land by the number of people. To do this, we first write each number in scientific notation.

\begin{gather*} 2,317,400,000 = 2.3174 \times 10^9\\ 1,335,300,000 = 1.3353 \times 10^9 \end{gather*}

We enter the division into the calculator as follows.

\begin{equation*} 2.3174~~\boxed{\text{EXP}}~~9~~\boxed{~\div ~}~~1.3353\boxed{\text{EXP}}~~9~~\boxed{~= ~} \end{equation*}

The calculator displays the quotient, 1.735. Thus, there are about 1.735 acres of land per person in China.

Subsection Mental Calculation with Scientific Notation

We can often obtain a quick estimate for a calculation by converting each figure to scientific notation and rounding to just one or two digits.

Example 9.22.

Use scientific notation to find the product

\begin{equation*} (6,200,000,000)(0.000 000 3) \end{equation*}
Solution

We can use a calculator (if necessary) to combine the decimal numbers, and the first law of exponents to combine the powers of 10.

\begin{align*} (6,200,000,000)(0.000 000 3) \amp = (6.2 \times 10^9)(3 \times 10^{-7})~~~~~~\blert{\text{Rearrange the factors.}}\\ \amp = (6.2 \times 3) \times (10^9 \times 10^{-7})~~~~~~ \blert{10^{9-7} = 10^2}\\ \amp = 18.6 \times 10^2 = 1860 \end{align*}

Subsection Skills Warm-Up

Exercises Exercises

Decide whether each statement is true or false.

1.
The reciprocal of \(x+2\) is \(\dfrac{1}{x} + \dfrac{1}{2}\text{.}\)
2.
The reciprocal of \(\dfrac{1}{3} + \dfrac{1}{4}\) is 7.
3.
\(\dfrac{3}{\dfrac{1}{5}} = 15\)
4.
\(\dfrac{3}{\dfrac{1}{5}+\dfrac{1}{2}} = 15+6\text{.}\)
5.
The reciprocal of \(\dfrac{1}{x}\) is \(x\text{.}\)
6.
The reciprocal of \(\dfrac{1}{x+y}\) is \(x+y\text{.}\)

Solutions Answers to Skills Warm-Up

Exercises Exercises

Exercises Homework 9.2

For Problems 1–2, write without using zero or negative exponents and simplify.

1.
  1. \(5^{-2}\)
  2. \(x^{-6}\)
  3. \((8x)^0\)
  4. \(\left(\dfrac{3}{4}\right)^{-3}\)
2.
  1. \(\left(\dfrac{b}{3}\right)^{-4}\)
  2. \((2q)^{-5}\)
  3. \(3 \cdot 4^{-3}\)
  4. \(4x^{-2}\)

For Problems 3–4, write each expression using negative exponents.

3.
  1. \(\dfrac{1}{2^3}\)
  2. \(\dfrac{3}{5^2}\)
  3. \(\dfrac{1}{27}\)
4.
  1. \(\dfrac{x}{625}\)
  2. \(\dfrac{2}{z^2}\)
  3. \(\left(\dfrac{z}{10}\right)^5\)
5.

Find and correct the error in each calculation.

  1. \(x^0 \rightarrow 0\)
  2. \(w^{-3} \rightarrow w^3\)
  3. \(2x^{-4} \rightarrow \dfrac{1}{2x^4}\)
6.

Simplify by using the first law of exponents.

  1. \(x^{-3} \cdot x^8\)
  2. \(5^{-4} \cdot 5^{-3}\)
  3. \((3b^{-5})(5b^2)\)
7.
Simplify by using the second law of exponents.
  1. \(\dfrac{c^{-7}}{c^{-4}}\)
  2. \(\dfrac{8b^{-4}}{4b^{-8}}\)
  3. \(\dfrac{6^6}{6^{-2}}\)
8.

Simplify.

  1. \(\dfrac{1}{6^{-3}}\)
  2. \(\dfrac{3}{2^{-6}}\)
  3. \(\dfrac{8x^3}{y^{-5}}\)
9.

Simplify by using the third law of exponents.

  1. \((8^{-2})^5\)
  2. \((w^{-6})^{-3}\)
  3. \((d^6)^{-4}\)
10.

Simplify by using the fourth law of exponents.

  1. \((pq)^{-5}\)
  2. \((3x)^{-2}\)
  3. \(5(2r)^{-3}\)

For Problems 11–14, simplify.

11.
\((a^{-4}c^2)^{-3}\)
12.
\((2u^2)^{-3}(u^{-4})^2\)
13.
\(\dfrac{5k^{-3}(k^4)^{-3}}{6k^{-5}}\)
14.
\(\left(\dfrac{2p^{-3}}{p^2}\right)^{-2}\)

Mental Exercise: For Problems 15–16, simplify each quotient without using pencil, paper, or calculator. Write your answer as a power of 10.

15.
  1. \(\dfrac{10^3}{10^{-2}}\)
  2. \(\dfrac{10^{-5}}{10^{-3}}\)
  3. \(\dfrac{10}{10^{-1}}\)
16.
  1. \(\dfrac{10^{-3}\times 10^{-2}}{10^{-6}}\)
  2. \(\dfrac{10^{-5}}{10^{-4} \times 10^7}\)
17.

Compute each product.

  1. \(4.3 \times 10^4\)
  2. \(8 \times 10^{-6}\)
  3. \(0.002 \times 10^{-2}\)
18.

Complete each factored form.

  1. \(234 = 2.34 \times \underline{\hspace{2.727272727272727em}}\)
  2. \(0.92 = 9.2 \times \underline{\hspace{2.727272727272727em}}\)
  3. \(1,720,000 = 1.72 \times \underline{\hspace{2.727272727272727em}}\)

For Problems 19–20, write in scientific notation.

19.

Compute each product.

  1. \(4834\)
  2. \(0.072\)
  3. \(0.000~007\)
20.

Complete each factored form.

  1. \(685,000,000\)
  2. \(56.74 \times 10^4\)
  3. \(385 \times 10^{-3}\)

For Problems 21–24, use scientific notation to compute.

21.
\((2,000,000)(0.000~07)\)
22.
\(0.000~036 \div 0.000~9\)
23.
\(\dfrac{(80,000,000,000)(0.000~6)}{20,000}\)
24.
\(\dfrac{(0.000~000~25)(90,000,000)}{(1,500,000)(0.000~000~2)}\)

For Problems 25–28, suppose that each of the following numbers is written in scientific notation. Decide whether the exponent on 10 is positive or negative.

25.

The population of Los Angeles County.

26.

The length of a football field in miles.

27.

The number of seconds in a year.

28.

The speed at which your hair grows in miles per hour.

29.

Write each number in scientific notation.

  1. The height of Mount Everest is 29,141 feet.
  2. The wavelength of red light is 0.000 076 centimeters.
30.

Write each number in standard notation.

  1. An amoeba weighs about \(5 \times 10^{-6}\) gram.
  2. Our solar system has existed for about \(5 \times 10^9\) years.

For Problems 31–34, give your answers in scientific notation rounded to two decimal places.

31.

A light year is the distance that light can travel in one year. Light travels at approximately \(1.86 \times 10^5\) miles per second. How many miles are there in one light year? (One year is approximately \(3.16 \times 10^7\) seconds.)

32.

A 1-foot high stack of dollar bills contains 3.610 bills.

  1. In January, 2018, the U. S. federal debt was \(2.152 \times 10^{13}\) dollars. How tall a stack of one-dollar bills, in feet, would be needed to pay off the federal debt?
  2. Express the height of the stack of bills in part (a) in miles. (One mile equals 5280 feet.)
33.

There are about 200 million insects for every person on earth. In 2018, the world's population is about 7.6 billion people.

  1. How many insects are there on earth?
  2. If the average insect weighs \(3\times 10^{-4}\) gram, how much do all the insects on earth weigh?
34.

There are 5,800,000 cubic miles of fresh water on earth. Each cubic mile is equal to 110,000,000,000 gallons of water. If the population of earth was about 7.6 billion people in 2018, how many gallons of fresh water were there for each person?