10.5 Integer Exponents and Scientific Notation

Learning objectives

Learning objectives

• Use the definition of a negative exponent

• Simplify expressions with integer exponents

• Convert from decimal notation to scientific notation

• Convert scientific notation to decimal form

• Multiply and divide using scientific notation

Use the Definition of a Negative Exponent

The Quotient Property of Exponents, introduced in Divide Monomials, had two forms depending on whether the exponent in the numerator or denominator was larger.

What if we just subtract exponents, regardless of which is larger? Let’s consider $\frac{{x}^{2}}{{x}^{5}}.$

We subtract the exponent in the denominator from the exponent in the numerator.

We can also simplify $\frac{{x}^{2}}{{x}^{5}}$ by dividing out common factors: $\frac{{x}^{2}}{{x}^{5}}.$

This implies that ${x}^{-3}=\frac{1}{{x}^{3}}$ and it leads us to the definition of a negative exponent.

The negative exponent tells us to 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 an expression with only positive exponents.

Example 10.63

Solution

1. Apply the negative-exponent definition. For a nonzero base, $a^{-n}=\frac{1}{a^n}$, so the base moves below the fraction bar.
2. Simplify part ⓐ. The positive power is $4^2=16$, so $4^{-2}=\frac{1}{16}$.
3. Simplify part ⓑ. The positive power is $10^3=1000$, so $10^{-3}=\frac{1}{1000}$.

Takeaway: A negative exponent means the reciprocal of the positive power, so $4^{-2}=\frac{1}{4^{2}}=\frac{1}{16}$ and $10^{-3}=\frac{1}{10^{3}}=\frac{1}{1000}$.

When simplifying any expression with exponents, we must be careful to correctly identify the base that is raised to each exponent.

Example 10.64

Solution

1. Identify the base in part ⓐ. The parentheses make the base $-3$, so the negative exponent requires the reciprocal of $(-3)^2$.
2. Simplify part ⓐ. Since $(-3)^2=9$, the result is $\frac{1}{9}$, which is positive.
3. Identify the base in part ⓑ. Without parentheses, the exponent applies only to $3$, while the leading negative sign remains outside.
4. Simplify part ⓑ. The expression becomes $-\frac{1}{3^2}=-\frac{1}{9}$, so the two answers have opposite signs.

Takeaway: Parentheses decide the base: $(-3)^{-2}=\frac{1}{9}$ because the base is $-3$, while $-3^{-2}=-\frac{1}{9}$ because the exponent applies only to $3$.

We must be careful to follow the order of operations. In the next example, parts ⓐ and ⓑ look similar, but we get different results.

Example 10.65

Solution

1. Use order of operations in part ⓐ. The exponent applies to $2$ before multiplication, so $4\cdot{}2^{-1}=4\cdot{}\frac{1}{2}=2$.
2. Simplify the parenthesized base in part ⓑ. First compute $4\cdot{}2=8$, so the expression becomes $(8)^{-1}$.
3. Apply the negative-exponent definition. The result is $\frac{1}{8}$, which shows that the exponent reaches different bases in the two parts.

Takeaway: Order of operations decides the answer: exponents come before multiplication, so $4\cdot2^{-1}=2$ but $(4\cdot2)^{-1}=\frac{1}{8}$.

When a variable is raised to a negative exponent, we apply the definition the same way we did with numbers.

Example 10.66

Solution

1. Identify the base and exponent. The base is $x$ and the exponent is $-6$, so the negative-exponent rule applies.
2. Rewrite the expression as a reciprocal. The definition $a^{-n}=\frac{1}{a^n}$ gives $x^{-6}=\frac{1}{x^6}$.
3. Check the base restriction. This form assumes $x\ne0$, and no further exponent operation is needed.

Takeaway: Using $a^{-n}=\frac{1}{a^{n}}$, $x^{-6}=\frac{1}{x^{6}}$: the negative exponent moves the base below the fraction bar without changing its sign.

When there is a product and an exponent we have to be careful to apply the exponent to the correct quantity. According to the order of operations, expressions in parentheses are simplified before exponents are applied. We’ll see how this works in the next example.

Example 10.67

Solution

1. Identify the base in part ⓐ. The exponent $-1$ applies only to $y$, so $5y^{-1}=5\cdot{}\frac{1}{y}=\frac{5}{y}$.
2. Identify the base in part ⓑ. Parentheses make $5y$ the base, so $(5y)^{-1}=\frac{1}{5y}$.
3. Identify the base in part ⓒ. The base is $-5y$, so taking its reciprocal gives $\frac{1}{-5y}=-\frac{1}{5y}$.
4. Compare the three forms. Each exponent acts only on the quantity directly beneath it, which changes which factors belong in the denominator.

Takeaway: The exponent applies only to what sits directly beneath it, so $5y^{-1}=\frac{5}{y}$, $(5y)^{-1}=\frac{1}{5y}$, and $(-5y)^{-1}=-\frac{1}{5y}$.

Now that we have defined negative exponents, the Quotient Property of Exponents needs only one form, $\frac{{a}^{m}}{{a}^{n}}={a}^{m-n},$ where $a\ne{}0$ and m and n are integers.

When the exponent in the denominator is larger than the exponent in the numerator, the exponent of the quotient will be negative. If the result gives us a negative exponent, we will rewrite it by using the definition of negative exponents, ${a}^{-n}=\frac{1}{{a}^{n}}.$

Simplify Expressions with Integer Exponents

All the exponent properties we developed earlier in this chapter with whole number exponents apply to integer exponents, too. We restate them here for reference.

Example 10.68

Solution

1. Use the product property in part ⓐ. The bases match, so add the exponents: $x^{-4}x^6=x^{-4+6}=x^2$.
2. Use the product property in part ⓑ. The exponent sum is $-6+4=-2$, so $y^{-6}y^4=y^{-2}=\frac{1}{y^2}$.
3. Use the product property in part ⓒ. Both exponents are negative, so their sum is $-5-3=-8$, giving $z^{-8}=\frac{1}{z^8}$.
4. Check each final form. Only part ⓐ has a positive resulting exponent, while the other two results are reciprocals.

Takeaway: For the same base, add the exponents and then clear any negative exponent: $x^{-4}x^{6}=x^{2}$, $y^{-6}y^{4}=\frac{1}{y^{2}}$, and $z^{-5}z^{-3}=\frac{1}{z^{8}}$.

In the next two examples, we’ll start by using the Commutative Property to group the same variables together. This makes it easier to identify the like bases before using the Product Property of Exponents.

Example 10.69

Solution

1. Regroup factors with like bases. Rearranging the product gives $m^4m^{-5}\cdot{}n^{-2}n^{-3}$.
2. Add the exponents for each base. The powers of $m$ give $m^{4-5}=m^{-1}$, and the powers of $n$ give $n^{-2-3}=n^{-5}$.
3. Rewrite both negative exponents. The definition of a negative exponent gives $m^{-1}n^{-5}=\frac{1}{m}\cdot\frac{1}{n^5}$.
4. State the simplified product. Combining the reciprocal factors gives the final result $\frac{1}{mn^5}$.

Takeaway: Regroup like bases before adding exponents: $(m^{4}n^{-3})(m^{-5}n^{-2})=m^{-1}n^{-5}=\frac{1}{mn^{5}}$.

If the monomials have numerical coefficients, we multiply the coefficients, just as we did in Use Multiplication Properties of Exponents.

Example 10.70

Solution

1. Regroup like bases and multiply coefficients. The expression becomes $2(-5)\cdot(x^{-6}x^5)\cdot(y^8y^{-3})$.
2. Simplify the coefficient. Multiplying $2$ and $-5$ gives $-10$, so the sign remains negative.
3. Add exponents for each variable. The powers of $x$ give $x^{-6+5}=x^{-1}$, and the powers of $y$ give $y^{8-3}=y^5$.
4. Rewrite the negative exponent. Replacing $x^{-1}$ with $\frac{1}{x}$ gives the final result $\frac{-10y^5}{x}$.

Takeaway: Multiply coefficients and add exponents base by base, then rewrite: $(2x^{-6}y^{8})(-5x^{5}y^{-3})=-10x^{-1}y^{5}=\frac{-10y^{5}}{x}$.

In the next two examples, we’ll use the Power Property and the Product to a Power Property.

Example 10.71

Solution

1. Apply the power property. A power raised to a power multiplies exponents, so ${(k^3)}^{-2}=k^{3(-2)}=k^{-6}$.
2. Rewrite the negative exponent. The definition of a negative exponent gives $k^{-6}=\frac{1}{k^6}$.
3. Check the sign in the exponent product. The product $3(-2)=-6$ is negative, so the reciprocal form is required.

Takeaway: A power of a power multiplies the exponents and a negative exponent flips to a reciprocal, so $(k^{3})^{-2}=k^{-6}=\frac{1}{k^{6}}$.

Example 10.72

Solution

1. Distribute the outer square. The power of a product property gives $(5x^{-3})^2=5^2(x^{-3})^2$.
2. Simplify the numerical power. Since $5^2=25$, the expression becomes $25(x^{-3})^2$.
3. Multiply the variable exponents. The power property gives $(x^{-3})^2=x^{-6}$, so the expression is $25x^{-6}$.
4. Rewrite the negative exponent. Using $x^{-6}=\frac{1}{x^6}$ produces the final result $\frac{25}{x^6}$.

Takeaway: The outside exponent applies to every factor: $(5x^{-3})^{2}=5^{2}x^{-6}=25x^{-6}=\frac{25}{x^{6}}$.

To simplify a fraction, we use the Quotient Property.

Example 10.73

Solution

1. Identify the common base. Both the numerator and denominator contain powers of $r$, so the quotient property applies: $\frac{{r}^{5}}{{r}^{-4}}=r^{5-(-4)}$.
2. Subtract the exponent carefully. The denominator exponent is negative, so $5-(-4)=5+4=9$. The second figure highlights the $-4$, and the third warns to perform the subtraction of a negative number correctly.
3. Write the simplified expression. The result is $r^9$, which is also displayed in the final figure. Therefore, $\frac{{r}^{5}}{{r}^{-4}}=r^9$.

Takeaway: Subtracting a negative exponent means adding its absolute value, a sign error that can change the entire result.

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 $4000$ and $0.004.$ We know that $4000$ means $4\,\times\,1000$ and $0.004$ means $4\,\times\,\frac{1}{1000}.$ If we write the $1000$ as a power of ten in exponential form, we can rewrite these numbers in this way:

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.

It is customary in scientific notation to use $\,\times\,$ as the multiplication sign, even though we avoid using this sign elsewhere in algebra.

Scientific notation is a useful way of writing very large or very small numbers. It is used often in the sciences to make calculations easier.

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

In both cases, the decimal was moved $3$ places to get the first factor, $4,$ by itself.

• The power of $10$ is positive when the number is larger than $1:\,4000=4\,\times\,{10}^{3}.$

• The power of $10$ is negative when the number is between $0$ and $1:\,0.004=4\,\times\,{10}^{-3}.$

Example 10.74

Solution

1. Move the decimal point. Place it after the first nonzero digit so that $37,000$ becomes $3.7$, a number between $1$ and $10$. The three trailing zeros in the original number mark the three places moved, as shown by the figure.
2. Count the decimal places. The decimal point moved three places to the left, so $n=3$.
3. Attach the power of ten. Because the original number is greater than $1$, write the product using $10^3$: $3.7\times10^3$.
4. Check the value. Moving the decimal point three places right in $3.7\times10^3$ gives $37,000$, so the scientific-notation form is $3.7\times10^3$.

Takeaway: Moving the decimal point left by $n$ places in a number greater than $1$ produces a factor of $10^n$.

Example 10.75

Solution

1. Move the decimal point to the first nonzero digit. Moving it three places right changes $0.0052$ into $5.2$, which lies between $1$ and $10$. The figure shows these three places with arrows.
2. Count the places moved. The decimal point moved three places to the right, so $n=3$.
3. Choose the power of ten. Because the original number is between $0$ and $1$, use $10^{-3}$ rather than $10^3$.
4. Check the value. The expression $5.2\times10^{-3}$ equals $5.2\div1000=0.0052$, so the scientific notation is $5.2\times10^{-3}$.

Takeaway: Moving the decimal point right by $n$ places in a number less than $1$ requires a negative exponent $-n$.

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.

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.

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 10.76

Solution

1. Read the exponent. The expression is $6.2\times10^3$, so the exponent $n$ is $3$.
2. Move the decimal point. Since the exponent is positive, move the decimal point three places to the right, changing $6.2$ to $6200$. The figure marks the three trailing positions that supply the zeros.
3. State the decimal form. The number $6.2\times10^3$ is $6200$ in decimal form.

Takeaway: A positive power of $10$ in scientific notation shifts the decimal point to the right and may require adding zeros.

Example 10.77

Solution

1. Interpret the negative exponent. Since $10^{-2}=1/10^2$, the expression is $8.9\div100$.
2. Move the decimal point two places left. Shifting $8.9$ to the left past two places gives $0.089$, so a zero must be written before the decimal point. The figure highlights the leading $8.9$ portion that is shifted.
3. State the decimal form. Thus $8.9\times10^{-2}=0.089$ in decimal form.

Takeaway: A negative exponent in scientific notation moves the decimal point left, and leading zeros must be retained.

Multiply and Divide Using Scientific Notation

We use the Properties of Exponents to multiply and divide numbers in scientific notation.

Example 10.78

Solution

1. Separate matching factors. Regroup the expression as $(4\cdot{}2)(10^5\cdot{}10^{-7})$.
2. Multiply the leading coefficients. The coefficient product is $4\cdot{}2=8$.
3. Add the exponents of ten. The exponent sum is $5+(-7)=-2$, giving $8\cdot{}10^{-2}$.
4. Convert to decimal form. A negative exponent of $-2$ moves the decimal two places left, so the result is $0.08$.

Takeaway: Multiply coefficients and add exponents separately: $(4\times10^{5})(2\times10^{-7})=8\times10^{-2}=0.08$.

Example 10.79

Solution

1. Separate the coefficient and powers of ten. Rewrite the quotient as $\frac{9}{3}\cdot\frac{10^3}{10^{-2}}$.
2. Divide the leading coefficients. The coefficient quotient is $\frac{9}{3}=3$.
3. Subtract the exponents of ten. Subtracting the denominator exponent gives $3-(-2)=5$, so the expression is $3\cdot{}10^5$.
4. Convert to decimal form. Moving the decimal five places right gives $300{,}000$.

Takeaway: Divide coefficients and subtract exponents: $\frac{9\times10^{3}}{3\times10^{-2}}=3\times10^{5}=300{,}000$.

Section 10.5 Exercises

Section 10.5 Exercises

Use the Definition of a Negative Exponent

In the following exercises, simplify.

Simplify Expressions with Integer Exponents

In the following exercises, simplify*.*

Convert from Decimal Notation to Scientific Notation

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

Convert Scientific Notation to Decimal Form

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

Multiply and Divide Using Scientific Notation

In the following exercises, multiply or divide and write your answer in decimal form.

Everyday Math

Writing Exercises

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After looking at the checklist, do you think you are well prepared for the next section? Why or why not?

1. Simplify:

1. ⓐ $\,{2}^{-3}$

2. ⓑ $\,{10}^{-2}$

2. Simplify:

1. ⓐ $\,{3}^{-2}$

2. ⓑ $\,{10}^{-4}$

3. Simplify:

1. ⓐ $\,{(-5)}^{-2}$

2. ⓑ $\,-{5}^{-2}$

4. Simplify:

1. ⓐ $\,{(-2)}^{-2}$

2. ⓑ $\,{-2}^{-2}$

5. Simplify:

1. ⓐ $\,6\cdot{3}^{-1}$

2. ⓑ $\,{(6\cdot{}3)}^{-1}$

1. Simplify:

1. ⓐ $\,8\cdot{2}^{-2}$

2. ⓑ $\,{(8\cdot{}2)}^{-2}$

2. Simplify: ${y}^{-7}.$

3. Simplify: ${z}^{-8}.$

4. Simplify:

1. ⓐ $\,8{p}^{-1}$

2. ⓑ $\,{(8p)}^{-1}$

3. ⓒ $\,{(-8p)}^{-1}$

5. Simplify:

1. ⓐ $\,11{q}^{-1}$

2. ⓑ $\,{(11q)}^{-1}$

3. ⓒ $\,{(-11q)}^{-1}$

1. Simplify:

1. ⓐ $\,{x}^{-3}\cdot{x}^{7}$

2. ⓑ $\,{y}^{-7}\cdot{y}^{2}$

3. ⓒ $\,{z}^{-4}\cdot{z}^{-5}$

2. Simplify:

1. ⓐ $\,{a}^{-1}\cdot{a}^{6}$

2. ⓑ $\,{b}^{-8}\cdot{b}^{4}$

3. ⓒ $\,{c}^{-8}\cdot{c}^{-7}$

3. Simplify: $({p}^{6}{q}^{-2})({p}^{-9}{q}^{-1}).$

4. Simplify: $({r}^{5}{s}^{-3})({r}^{-7}{s}^{-5}).$

5. Simplify: $(3{u}^{-5}{v}^{7})(-4{u}^{4}{v}^{-2}).$

1. Simplify: $(-6{c}^{-6}{d}^{4})(-5{c}^{-2}{d}^{-1}).$

2. Simplify: ${({x}^{4})}^{-1}.$

3. Simplify: ${({y}^{2})}^{-2}.$

4. Simplify: ${(8{a}^{-4})}^{2}.$

5. Simplify: ${(2{c}^{-4})}^{3}.$

1. Simplify: $\frac{{x}^{8}}{{x}^{-3}}.$

2. Simplify: $\frac{{y}^{7}}{{y}^{-6}}.$

3. Write in scientific notation: $96,000.$

4. Write in scientific notation: $48,300.$

5. Write in scientific notation: $0.0078.$

1. Write in scientific notation: $0.0129.$

2. Convert to decimal form: $1.3\,\times\,{10}^{3}.$

3. Convert to decimal form: $9.25\,\times\,{10}^{4}.$

4. Divide. Write answers in decimal form: $\frac{8\,\times\,{10}^{4}}{2\,\times\,{10}^{-1}}.$

5. Divide. Write answers in decimal form: $\frac{8\,\times\,{10}^{2}}{4\,\times\,{10}^{-2}}.$

1. ${8}^{-2}$

2. ${2}^{-5}$

3. ${10}^{-1}$

4. ${3}^{-2}+{3}^{-1}$

5. ${10}^{-1}+{2}^{-1}$

1. ${2}^{0}-{2}^{-1}+{2}^{-2}$

2. 1. ⓐ $\,{(-8)}^{-2}$

2. ⓑ $\,-{8}^{-2}$

3. 1. ⓐ $\,{(-4)}^{-6}$

2. ⓑ $\,-{4}^{-6}$

4. 1. ⓐ $\,10\cdot{3}^{-1}$

2. ⓑ $\,{(10\cdot{}3)}^{-1}$

5. 1. ⓐ $\,3\cdot{5}^{-2}$

2. ⓑ $\,{(3\cdot{}5)}^{-2}$

1. ${p}^{-3}$

2. ${m}^{-5}$

3. 1. ⓐ $\,3{q}^{-1}$

2. ⓑ $\,{(3q)}^{-1}$

3. ⓒ $\,{(-3q)}^{-1}$

4. 1. ⓐ $\,10{k}^{-1}$

2. ⓑ $\,{(10k)}^{-1}$

3. ⓒ $\,{(-10k)}^{-1}$

5. ${r}^{-2}\cdot{r}^{5}$

1. ${q}^{-8}\cdot{q}^{3}$

2. ${z}^{-6}\cdot{z}^{-2}$

3. $m\cdot{m}^{-2}$

4. ${x}^{4}\cdot{x}^{-2}\cdot{x}^{-3}$

5. ${u}^{2}{v}^{-2}$

1. $({a}^{3}{b}^{-3})({a}^{-5}{b}^{-1})$

2. $(p{q}^{-4})({p}^{-6}{q}^{-3})$

3. $(-3{p}^{-5}{q}^{8})(7{p}^{2}{q}^{-3})$

4. $(-8{a}^{-5}{b}^{-4})(-4{a}^{2}{b}^{3})$

5. ${({q}^{10})}^{-10}$

1. ${({x}^{4})}^{-1}$

2. ${({p}^{-3})}^{2}$

3. ${({m}^{-2})}^{-3}$

4. ${(3{q}^{-5})}^{2}$

5. ${(2{n}^{-3})}^{-6}$

1. $\frac{{b}^{5}}{{b}^{-3}}$

2. $\frac{{m}^{5}}{{m}^{-2}}$

3. $\frac{{r}^{6}}{{r}^{9}}$

4. $\frac{{p}^{-3}}{{p}^{-6}}$

5. 280,000

1. 1,290,000

2. The population of the world on July 4, 2010 was more than $6,850,000,000.$

3. The probability of winning the $2010$ Megamillions lottery is about $0.0000000057.$

4. $1.6\,\times\,{10}^{10}$

5. At the start of 2012, the US federal budget had a deficit of more than $\text{\textdollar 1.5}\,\times\,{10}^{13}.$

1. $\frac{8\,\times\,{10}^{6}}{4\,\times\,{10}^{-1}}$

2. $\frac{5\,\times\,{10}^{-3}}{1\,\times\,{10}^{-10}}$

3. Length of a year The difference between the calendar year and the astronomical year is $0.000125$ day.

1. ⓐ Write this number in scientific notation.

2. ⓑ How many years does it take for the difference to become 1 day?

4. Calculator display Many calculators automatically show answers in scientific notation if there are more digits than can fit in the calculator’s display. To find the number of ways Barbara could make a collage with $6$ of her $50$ favorite photographs, she multiplied $50\cdot{}49\cdot{}48\cdot{}47\cdot{}46\cdot{}45.$ Her calculator gave the answer $1.1441304\,\times\,{10}^{10}.$ Write the number in decimal notation.

5. When you convert a number from decimal notation to scientific notation, how do you know if the exponent will be positive or negative?