Integer Exponents and Scientific Notation

By the end of this section, you will be able to: 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, and multiply and divide using scientific notation.

Use the Definition of a Negative Exponent

The Quotient Property of Exponents had two forms depending on whether the exponent in the numerator or denominator was larger:

Quotient Property of Exponents. If $a$ is a real number, $a \neq 0$ , and $m, n$ are whole numbers, then

\frac{a^m}{a^n} = a^{m-n}, \quad m > n \qquad\text{and}\qquad \frac{a^m}{a^n} = \frac{1}{a^{n-m}}, \quad n > m

What if we just subtract exponents, regardless of which is larger? Let’s consider $\tfrac{x^2}{x^5}$ . We subtract the exponent in the denominator from the exponent in the numerator:

\frac{x^2}{x^5} = x^{2-5} = x^{-3}

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

\frac{x \cdot x}{x \cdot x \cdot x \cdot x \cdot x} = \frac{1}{x^3}

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

Negative Exponent. If $n$ is a positive integer and $a \neq 0$ , then

a^{-n} = \frac{1}{a^n}

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

(a) Use the definition of a negative exponent, $a^{-n} = \tfrac{1}{a^n}$ :

4^{-2} = \frac{1}{4^2} = \frac{1}{16}

(b) Use the same definition:

10^{-3} = \frac{1}{10^3} = \frac{1}{1000}

Simplify: $2^{-3}$ .

Simplify: $10^{-2}$ .

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

Example. Simplify: (a) $(-3)^{-2}$ (b) $-3^{-2}$ . The negative in the exponent does not affect the sign of the base.

(a) The exponent applies to the base, $-3$ :

(-3)^{-2} = \frac{1}{(-3)^2} = \frac{1}{9}

(b) The expression $-3^{-2}$ means “find the opposite of $3^{-2}$ .” The exponent applies only to the base, $3$ . Rewrite as a product with $-1$ :

-3^{-2} = -1 \cdot 3^{-2} = -1 \cdot \frac{1}{3^2} = -\frac{1}{9}

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

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

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

Example. Simplify: (a) $4 \cdot 2^{-1}$ (b) $(4 \cdot 2)^{-1}$ . Remember to always follow the order of operations.

(a) Do exponents before multiplication:

4 \cdot 2^{-1} = 4 \cdot \frac{1}{2^1} = 2

(b) Simplify inside the parentheses first:

(4 \cdot 2)^{-1} = (8)^{-1} = \frac{1}{8^1} = \frac{1}{8}

Simplify: $6 \cdot 3^{-1}$ .

Simplify: $(6 \cdot 3)^{-1}$ .

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

Example. Simplify: $x^{-6}$ .

Use the definition of a negative exponent, $a^{-n} = \tfrac{1}{a^n}$ :

x^{-6} = \frac{1}{x^6}

Simplify: $y^{-7}$ .

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.

Example. Simplify: (a) $5y^{-1}$ (b) $(5y)^{-1}$ (c) $(-5y)^{-1}$ .

(a) Notice the exponent applies to just the base $y$ . Take the reciprocal of $y$ and change the sign of the exponent:

5y^{-1} = 5 \cdot \frac{1}{y^1} = \frac{5}{y}

(b) Here the parentheses make the exponent apply to the base $5y$ . Take the reciprocal of $5y$ and change the sign of the exponent:

(5y)^{-1} = \frac{1}{(5y)^1} = \frac{1}{5y}

(c) The base is $-5y$ . Take the reciprocal of $-5y$ and change the sign of the exponent, then use $\tfrac{a}{-b} = -\tfrac{a}{b}$ :

(-5y)^{-1} = \frac{1}{(-5y)^1} = \frac{1}{-5y} = -\frac{1}{5y}

Simplify: $8p^{-1}$ .

Simplify: $(8p)^{-1}$ .

Simplify: $(-8p)^{-1}$ .

Now that we have defined negative exponents, the Quotient Property of Exponents needs only one form, $\tfrac{a^m}{a^n} = a^{m-n}$ , where $a \neq 0$ 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 rewrite it using the definition of negative exponents, $a^{-n} = \tfrac{1}{a^n}$ .

Simplify Expressions with Integer Exponents

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

Summary of Exponent Properties. If $a, b$ are real numbers and $m, n$ are integers, then

\text{Product Property} \qquad a^m \cdot a^n = a^{m+n}

\text{Power Property} \qquad (a^m)^n = a^{m \cdot n}

\text{Product to a Power Property} \qquad (ab)^m = a^m b^m

\text{Quotient Property} \qquad \frac{a^m}{a^n} = a^{m-n}, \quad a \neq 0

\text{Zero Exponent Property} \qquad a^0 = 1, \quad a \neq 0

\text{Quotient to a Power Property} \qquad \left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}, \quad b \neq 0

\text{Definition of Negative Exponent} \qquad a^{-n} = \frac{1}{a^n}

Example. Simplify: (a) $x^{-4} \cdot x^6$ (b) $y^{-6} \cdot y^4$ (c) $z^{-5} \cdot z^{-3}$ .

(a) Use the Product Property, $a^m \cdot a^n = a^{m+n}$ :

x^{-4} \cdot x^6 = x^{-4+6} = x^2

(b) The bases are the same, so add the exponents:

y^{-6} \cdot y^4 = y^{-6+4} = y^{-2} = \frac{1}{y^2}

z^{-5} \cdot z^{-3} = z^{-5-3} = z^{-8} = \frac{1}{z^8}

Simplify: $x^{-3} \cdot x^7$ .

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

In the next two examples, we 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. Simplify: $(m^4 n^{-3})(m^{-5} n^{-2})$ .

Use the Commutative Property to get like bases together, then add the exponents for each base:

(m^4 n^{-3})(m^{-5} n^{-2}) = m^4 m^{-5} \cdot n^{-3} n^{-2} = m^{-1} \cdot n^{-5} = \frac{1}{m^1} \cdot \frac{1}{n^5} = \frac{1}{mn^5}

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

If the monomials have numerical coefficients, we multiply the coefficients, just as we did earlier in the chapter.

Example. Simplify: $(2x^{-6} y^8)(-5x^5 y^{-3})$ .

Rewrite with the like bases together, multiply the coefficients, add the exponents for each base, and rewrite with only positive exponents:

(2x^{-6} y^8)(-5x^5 y^{-3}) = 2(-5) \cdot (x^{-6} x^5) \cdot (y^8 y^{-3}) = -10 \cdot x^{-1} \cdot y^5 = -10 \cdot \frac{1}{x^1} \cdot y^5 = \frac{-10y^5}{x}

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

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

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

Use the Power Property, $(a^m)^n = a^{m \cdot n}$ , then rewrite with a positive exponent:

(k^3)^{-2} = k^{3(-2)} = k^{-6} = \frac{1}{k^6}

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

Example. Simplify: $(5x^{-3})^2$ .

Use the Product to a Power Property, $(ab)^m = a^m b^m$ , then simplify $5^2$ and multiply the exponents of $x$ using the Power Property, and finally rewrite $x^{-6}$ using the definition of a negative exponent:

(5x^{-3})^2 = 5^2 (x^{-3})^2 = 25 x^{-6} = 25 \cdot \frac{1}{x^6} = \frac{25}{x^6}

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

To simplify a fraction, we use the Quotient Property.

Example. Simplify: $\tfrac{r^5}{r^{-4}}$ .

Use the Quotient Property, $\tfrac{a^m}{a^n} = a^{m-n}$ , being careful to subtract $5 - (-4)$ :

\frac{r^5}{r^{-4}} = r^{5-(-4)} = r^9

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

Convert from Decimal Notation to Scientific Notation

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 ten — 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 $4 \times \tfrac{1}{1000}$ . If we write the $1000$ as a power of ten in exponential form, we can rewrite these numbers this way:

4000 = 4 \times 1000 = 4 \times 10^3 \qquad\qquad 0.004 = 4 \times \frac{1}{1000} = 4 \times 10^{-3}

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

a \times 10^n

where $a \geq 1$ and $a < 10$ and $n$ is an integer.

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. Moving the decimal point three places to the left in $4000$ , and three places to the right in $0.004$ , gets the first factor, $4$ , by itself in both cases:

4000. = 4 \times 10^3 \qquad\qquad 0.004 = 4 \times 10^{-3}

In both cases, the decimal was moved three places to get the first factor 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 $1$ : $0.004 = 4 \times 10^{-3}$ .

Convert from decimal notation to scientific notation.

Move the decimal point so that the first factor is greater than or equal to $1$ but less than $10$ .
Count the number of decimal places, $n$ , that the decimal point was moved.
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 $10^n$ ; if the original number is between $0$ and $1$ , the power of $10$ will be $10^{-n}$ .
Check.

Example. Write $37{,}000$ in scientific notation.

Move the decimal point so that the first factor is greater than or equal to $1$ but less than $10$ ; count that the decimal point moved $4$ places; then write the number as a product with a power of $10$ :

37{,}000 = 3.7 \times 10^4

Check: $10^4$ is $10{,}000$ , and $10{,}000$ times $3.7$ is $37{,}000$ . ✓

Write in scientific notation: 96,000.

Example. Write $0.0052$ in scientific notation.

Move the decimal point to get $5.2$ , a number between $1$ and $10$ ; count that the point moved $3$ places; then write as a product with a power of $10$ . Since the original number is between $0$ and $1$ , the exponent is negative:

0.0052 = 5.2 \times 10^{-3}

Check: $5.2 \times 10^{-3} = 5.2 \times \tfrac{1}{1000} = 5.2 \times 0.001 = 0.0052$ . ✓

Write in scientific notation: 0.0078.

Convert Scientific Notation to Decimal Form

To convert scientific notation to decimal form, look at two numbers written in scientific notation and see what happens to the decimal point:

9.12 \times 10^4 = 91{,}200 \qquad\qquad 9.12 \times 10^{-4} = 0.000912

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 moved to the left.

Convert scientific notation to decimal form.

Determine the exponent, $n$ , on the factor $10$ .
Move the decimal $n$ places, adding zeros if needed — to the right if the exponent is positive, to the left if the exponent is negative.
Check.

Example. Convert to decimal form: $6.2 \times 10^3$ .

The exponent is $3$ , so move the decimal point $3$ places to the right, adding zeros if needed:

6.2 \times 10^3 = 6{,}200

Check: $10^3$ is $1000$ , and $1000$ times $6.2$ is $6{,}200$ . ✓

Convert to decimal form: $1.3 \times 10^3$ .

Example. Convert to decimal form: $8.9 \times 10^{-2}$ .

The exponent is $-2$ , so move the decimal point $2$ places to the left, adding zeros as needed for placeholders:

8.9 \times 10^{-2} = 0.089

Convert to decimal form: $1.2 \times 10^{-4}$ .

Multiply and Divide Using Scientific Notation

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

Example. Multiply. Write the answer in decimal form: $(4 \times 10^5)(2 \times 10^{-7})$ .

Use the Commutative Property to rearrange the factors, multiply $4$ by $2$ and use the Product Property to multiply $10^5$ by $10^{-7}$ , then change to decimal form by moving the decimal two places left:

(4 \times 10^5)(2 \times 10^{-7}) = 4 \cdot 2 \cdot 10^5 \cdot 10^{-7} = 8 \times 10^{-2} = 0.08

Multiply. Write the answer in decimal form: $(3 \times 10^6)(2 \times 10^{-8})$ .

Example. Divide. Write the answer in decimal form: $\tfrac{9 \times 10^3}{3 \times 10^{-2}}$ .

Separate the factors, divide $9$ by $3$ and use the Quotient Property to divide $10^3$ by $10^{-2}$ , then change to decimal form by moving the decimal five places right:

\frac{9 \times 10^3}{3 \times 10^{-2}} = \frac{9}{3} \times \frac{10^3}{10^{-2}} = 3 \times 10^5 = 300{,}000

Divide. Write the answer in decimal form: $\tfrac{8 \times 10^4}{2 \times 10^{-1}}$ .

Key terms

negative exponent — for a positive integer $n$ and $a \neq 0$ , $a^{-n} = \tfrac{1}{a^n}$ ; the exponent tells us to take the reciprocal of the base and change the sign of the exponent. scientific notation — a number written as $a \times 10^n$ , where $1 \le a < 10$ and $n$ is an integer, used to conveniently express very large or very small numbers.

This section is adapted from Prealgebra 2e, Section 10.5: Integer Exponents and Scientific Notation by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: omitted the Be Prepared quiz, Media links, Self Check checklist, and end-of-section exercises; and converted the practice problems (“Try Its”) into interactive exercises with instant feedback.

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