10.8: Integer Exponents and Scientific Notation (Part 1)

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Skills to Develop

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

be prepared!

Before you get started, take this readiness quiz.

What is the place value of the 6 in the number 64,891? If you missed this problem, review Example 1.1.3.
Name the decimal 0.0012. If you missed this problem, review Exercise 5.1.1.
Subtract: 5 − (−3). If you missed this problem, review Example 3.5.8.

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.

Definition: Quotient Property of Exponents

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

\[\dfrac{a^{m}}{a^{n}} = a^{m-n},\; m>n \quad and \quad \dfrac{a^{m}}{a^{n}} = \dfrac{1}{a^{n-m}},\; n>m\]

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

\[\begin{split} &\; \dfrac{x^{2}}{x^{5}} \\ &x^{2-5} \\ &x^{-3} \end{split}\]

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

\[\begin{split} &\dfrac{\cancel{x} \cdot \cancel{x}}{\cancel{x} \cdot \cancel{x} \cdot x \cdot x \cdot x} \\ &\qquad \quad \dfrac{1}{x^{3}} \end{split}\]

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

Definition: negative exponent

If n is a positive integer and a ≠ 0, then $a^{−n} = \dfrac{1}{a^{n}}$ .

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 $\PageIndex{1}$:

Simplify: (a) 4⁻² (b) 10⁻³

Solution

(a) 4⁻²

Use the definition of a negative exponent, $a^{−n} = \dfrac{1}{a^{n}}$.	$$\dfrac{1}{4^{2}}$$
Simplify.	$$\dfrac{1}{16}$$

(b) 10⁻³

Use the definition of a negative exponent, $a^{−n} = \dfrac{1}{a^{n}}$.	$$\dfrac{1}{10^{3}}$$
Simplify.	$$\dfrac{1}{1000}$$

Exercise $\PageIndex{1}$:

Simplify: (a) 2⁻³ (b) 10⁻²

Answer a: $\frac{1}{8}$

Answer b: $\frac{1}{100}$

Exercise $\PageIndex{2}$:

Simplify: (a) 3⁻² (b) 10⁻⁴

Answer a: $\frac{1}{9}$

Answer b: $\frac{1}{10,000}$

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

Example $\PageIndex{2}$:

Simplify: (a) (−3)⁻² (b) −3⁻²

Solution

The negative in the exponent does not affect the sign of the base.

a) (−3)⁻²

The exponent applies to the base, −3.	$$(-3)^{-2}$$
Take the reciprocal of the base and change the sign of the exponent.	$$\dfrac{1}{(-3)^{2}}$$
Simplify.	$$\dfrac{1}{9}$$

(b) −3⁻²

The expression −3⁻² means "find the opposite of 3⁻². The exponent applies only to the base, 3.	$$-3^{-2}$$
Rewrite as a product with −1.	$$-1 \cdot 3^{-2}$$
Take the reciprocal of the base and change the sign of the exponent.	$$-1 \cdot \dfrac{1}{3^{2}}$$
Simplify.	$$- \dfrac{1}{9}$$

Exercise $\PageIndex{3}$:

Simplify: (a) (−5)⁻² (b) −5⁻²

Answer a: $\frac{1}{25}$

Answer b: $-\frac{1}{25}$

Exercise $\PageIndex{4}$:

Simplify: (a) (−2)⁻² (b) −2⁻²

Answer a: $\frac{1}{4}$

Answer b: $-\frac{1}{4}$

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 $\PageIndex{3}$:

Simplify: (a) 4 • 2⁻¹ (b) (4 • 2)⁻¹

Solution

Remember to always follow the order of operations.

(a) 4 • 2⁻¹

Do exponents before multiplication.	$$4 \cdot 2^{-1}$$
Use $a^{−n} = \dfrac{1}{a^{n}}$.	$$4 \cdot \dfrac{1}{2^{1}}$$
Simplify.	$$2$$

(b) (4 • 2)⁻¹

Simplify inside the parentheses first.	$$(8)^{-1}$$
Use $a^{−n} = \dfrac{1}{a^{n}}$.	$$\dfrac{1}{8^{1}}$$
Simplify.	$$\dfrac{1}{8}$$

Exercise $\PageIndex{5}$:

Simplify: (a) 6 • 3⁻¹ (b) (6 • 3)⁻¹

Answer a: $2$

Answer b: $\frac{1}{18}$

Exercise $\PageIndex{6}$:

Simplify: (a) 8 • 2⁻² (b) (8 • 2)⁻²

Answer a: $2$

Answer b: $\frac{1}{256}$

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

Example $\PageIndex{4}$:

Simplify: x⁻⁶.

Solution

Use the definition of a negative exponent, $a^{−n} = \dfrac{1}{a^{n}}$.

$$\dfrac{1}{x^{6}}$$

Exercise $\PageIndex{7}$:

Simplify: y⁻⁷.

Answer: $\frac{1}{y^7}$

Exercise $\PageIndex{8}$:

Simplify: z^-8.

Answer: $\frac{1}{z^8}$

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 $\PageIndex{5}$:

Simplify: (a) 5y⁻¹ (b) (5y)⁻¹ (c) (−5y)⁻¹

Solution

(a) 5y⁻¹

Notice the exponent applies to just the base y .	$$5y^{-1}$$
Take the reciprocal of y and change the sign of the exponent.	$$5 \cdot \dfrac{1}{y^{1}}$$
Simplify.	$$\dfrac{5}{y}$$

(b) (5y)⁻¹

Here the parentheses make the exponent apply to the base 5y.	$$(5y)^{-1}$$
Take the reciprocal of 5y and change the sign of the exponent.	$$\dfrac{1}{(5y)^{1}}$$
Simplify.	$$\dfrac{1}{5y}$$

The base is −5y . Take the reciprocal of −5y and change the sign of the exponent.	$$\dfrac{1}{(-5y)^{1}}$$
Simplify.	$$\dfrac{1}{-5y}$$
Use $\dfrac{a}{-b} = - \dfrac{a}{b}$.	$$- \dfrac{1}{5y}$$

Exercise $\PageIndex{9}$:

Simplify: (a) 8p⁻¹ (b) (8p)⁻¹ (c) (−8p)⁻¹

Answer a: $\frac{8}{p}$

Answer b: $\frac{1}{8p}$

Answer c: $-\frac{1}{8p}$

Exercise $\PageIndex{10}$:

Simplify: (a) 11q⁻¹ (b) (11q)⁻¹ (c) (−11q)⁻¹

Answer a: $\frac{11}{q}$

Answer b: $\frac{1}{11q}$

Answer c: $-\frac{1}{11q}$

Now that we have defined negative exponents, the Quotient Property of Exponents needs only one form, $\dfrac{a^{m}}{a^{n}} = a^{m − n}$, where a ≠ 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, $a^{−n} = \dfrac{1}{a^{n}}$.

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.

Summary of Exponent Properties

If a, b are real numbers and m, n are integers, then

Product Property	a^m • aⁿ = a^{m + n}
Power Property	(a^m)ⁿ = a^{m • n}
Product to a Power Property	(ab)^m = a^mb^m
Quotient Property	$\dfrac{a^{m}}{a^{n}}$ = a^{m − n}, a ≠ 0, m > n
	$\dfrac{a^{m}}{a^{n}} = \dfrac{1}{a^{n-m}}$, a ≠ 0, n > m
Zero Exponent Property	a⁰ = 1, a ≠ 0
Quotient to a Power Property	$\left(\dfrac{a}{b}\right)^{m} = \dfrac{a^{m}}{b^{m}}$, b ≠ 0
Definition of a Negative Exponent	$a^{-n} = \dfrac{1}{a^{n}}$

Example $\PageIndex{6}$:

Simplify: (a) x⁻⁴ • x⁶ (b) y⁻⁶ • y⁴ (c) z⁻⁵• z⁻³

Solution

(a) x⁻⁴ • x⁶

Use the Product Property, a^m • aⁿ = a^{m + n}.	$$x^{-4+6}$$
Simplify.	$$x^{2}$$

(b) y⁻⁶ • y⁴

The bases are the same, so add the exponents.	$$y^{-6+4}$$
Simplify.	$$y^{-2}$$
Use the definition of a negative exponent, $a^{−n} = \dfrac{1}{a^{n}}$.	$$\dfrac{1}{y^{2}}$$

The bases are the same, so add the exponents.	$$z^{-5-3}$$
Simplify.	$$z^{-8}$$
Use the definition of a negative exponent, $a^{−n} = \dfrac{1}{a^{n}}$.	$$\dfrac{1}{z^{8}}$$

Exercise $\PageIndex{11}$:

Simplify: (a) x⁻³ • x⁷ (b) y⁻⁷ • y² (c) z⁻⁴• z⁻⁵

Answer a: $x^4$

Answer b: $\frac{1}{y^5}$

Answer c: $\frac{1}{z^9}$

Exercise $\PageIndex{12}$:

Simplify: (a) a⁻¹ • a⁶ (b) b⁻⁶ • b⁴ (c) c⁻⁸• c⁻⁷

Answer a: $a^5$

Answer b: $\frac{1}{b^4}$

Answer c: $\frac{1}{c^{15}}$

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 $\PageIndex{7}$:

Simplify: (m⁴n⁻³)(m⁻⁵n⁻²).

Solution

Use the Commutative Property to get like bases together.	$$m^{4} m^{-5} \cdot n^{-2} n^{-3}$$
Add the exponents for each base.	$$m^{-1} \cdot n^{-5}$$
Take reciprocals and change the signs of the exponents.	$$\dfrac{1}{m^{1}} \cdot \dfrac{1}{n^{5}}$$
Simplify.	$$\dfrac{1}{mn^{5}}$$

Exercise $\PageIndex{13}$:

Simplify: (p⁶q⁻²)(p⁻⁹q⁻¹).

Answer: $\frac{1}{p^3q^3}$

Exercise $\PageIndex{14}$:

Simplify: (r⁵s⁻³)(r⁻⁷s⁻⁵).

Answer: $\frac{1}{r^2 s^8}$

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

Example $\PageIndex{8}$:

Simplify: (2x⁻⁶y⁸)(−5x⁵ y⁻³).

Solution

Rewrite with the like bases together.	$$2(-5) \cdot (x^{-6} x^{5}) \cdot (y^{8} y^{-3})$$
Simplify.	$$-10 \cdot x^{-1} \cdot y^{5}$$
Use the definition of a negative exponent, $a^{−n} = \dfrac{1}{a^{n}}$.	$$-10 \cdot \dfrac{1}{x^{1}} \cdot y^{5}$$
Simplify.	$$\dfrac{-10y^{5}}{x}$$

Exercise $\PageIndex{15}$:

Simplify: (3u⁻⁵v⁷)(−4u⁴v⁻²).

Answer: $-\frac{12v^5}{u}$

Exercise $\PageIndex{16}$:

Simplify: (−6c⁻⁶d⁴)(−5c⁻²d⁻¹).

Answer: $\frac{30d^3}{c^8}$

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

Example $\PageIndex{9}$:

Simplify: (k³)⁻².

Solution

Use the Product to a Power Property, (ab)^m = a^mb^m.	$$k^{3(-2)}$$
Simplify.	$$k^{-6}$$
Rewrite with a positive exponent.	$$\dfrac{1}{k^{6}}$$

Exercise $\PageIndex{17}$:

Simplify: (x⁴)⁻¹.

Answer: $\frac{1}{x^4}$

Exercise $\PageIndex{18}$:

Simplify: (y²)⁻².

Answer: $\frac{1}{y^4}$

Example $\PageIndex{10}$:

Simplify: (5x⁻³)².

Solution

Use the Product to a Power Property, (ab)^m = a^mb^m.	$$5^{2} (x^{-3})^{2}$$
Simplify 5² and multiply the exponents of x using the Power Property, (a^m)ⁿ = a^{m • n}.	$$25k^{-6}$$
Rewrite x⁻⁶ by using the definition of a negative exponent, $a^{−n} = \dfrac{1}{a^{n}}$.	$$25 \cdot \dfrac{1}{x^{6}}$$
Simplify.	$$\dfrac{25}{x^{6}}$$

Exercise $\PageIndex{19}$:

Simplify: (8a⁻⁴)².

Answer: $\frac{64}{a^8}$

Exercise $\PageIndex{20}$:

Simplify: (2c⁻⁴)³.

Answer: $\frac{8}{c^12}$

To simplify a fraction, we use the Quotient Property.

Example $\PageIndex{11}$:

Simplify: $\dfrac{r^{5}}{r^{−4}}$.

Solution

Use the Quotient Property, $\dfrac{a^{m}}{a^{n}} = a^{m-n}$.	$$r^{5-(\textcolor{red}{-4})}$$
Be careful to subtract 5 - (\textcolor{red}{-4}).
Simplify.	$$r^{9}$$

Exercise $\PageIndex{21}$:

Simplify: $\dfrac{x^{8}}{x^{−3}}$.

Answer: $x^{11}$

Exercise $\PageIndex{22}$:

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

Answer: $y^{13}$

Contributors and Attributions

Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (Formerly of Santa Ana College). This content is licensed under Creative Commons Attribution License v4.0 "Download for free at http://cnx.org/contents/fd53eae1-fa2...49835c3c@5.191."

Do exponents before multiplication.	$$4 \cdot 2^{-1}$$
Use \(a^{−n} = \dfrac{1}{a^{n}}\).	$$4 \cdot \dfrac{1}{2^{1}}$$
Simplify.	$$2$$

Use the Quotient Property, \(\dfrac{a^{m}}{a^{n}} = a^{m-n}\).	$$r^{5-(\textcolor{red}{-4})}$$
Be careful to subtract 5 - (\textcolor{red}{-4}).
Simplify.	$$r^{9}$$