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10.8: Integer Exponents and Scientific Notation (Part 1)

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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
be prepared!

Before you get started, take this readiness quiz.

  1. What is the place value of the 6 in the number 64,891? If you missed this problem, review Example 1.1.3.
  2. Name the decimal 0.0012. If you missed this problem, review Exercise 5.1.1.
  3. 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

aman=amn,m>nandaman=1anm,n>m

What if we just subtract exponents, regardless of which is larger? Let’s consider x2x5. We subtract the exponent in the denominator from the exponent in the numerator.

x2x5x25x3

We can also simplify x2x5 by dividing out common factors: x2x5.

xxxxxxx1x3

This implies that x3=1x3 and it leads us to the definition of a negative exponent.

Definition: negative exponent

If n is a positive integer and a ≠ 0, then an=1an.

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 10.8.1:

Simplify: (a) 4−2 (b) 10−3

Solution

(a) 4−2

Use the definition of a negative exponent, an=1an. 142
Simplify. 116

(b) 10−3

Use the definition of a negative exponent, an=1an. 1103
Simplify. 11000
Exercise 10.8.1:

Simplify: (a) 2−3 (b) 10−2

Answer a

18

Answer b

1100

Exercise 10.8.2:

Simplify: (a) 3−2 (b) 10−4

Answer a

19

Answer b

110,000

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

Example 10.8.2:

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. 1(3)2
Simplify. 19

(b) −3−2

The expression −3−2 means "find the opposite of 3−2. The exponent applies only to the base, 3. 32
Rewrite as a product with −1. 132
Take the reciprocal of the base and change the sign of the exponent. 1132
Simplify. 19
Exercise 10.8.3:

Simplify: (a) (−5)−2 (b) −5−2

Answer a

125

Answer b

125

Exercise 10.8.4:

Simplify: (a) (−2)−2 (b) −2−2

Answer a

14

Answer b

14

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 10.8.3:

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. 421
Use an=1an. 4121
Simplify. 2

(b) (4 • 2)−1

Simplify inside the parentheses first. (8)1
Use an=1an. 181
Simplify. 18
Exercise 10.8.5:

Simplify: (a) 6 • 3−1 (b) (6 • 3)−1

Answer a

2

Answer b

118

Exercise 10.8.6:

Simplify: (a) 8 • 2−2 (b) (8 • 2)−2

Answer a

2

Answer b

1256

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

Example 10.8.4:

Simplify: x−6.

Solution

Use the definition of a negative exponent, an=1an. 1x6
Exercise 10.8.7:

Simplify: y−7.

Answer

1y7

Exercise 10.8.8:

Simplify: z-8.

Answer

1z8

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 10.8.5:

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

Solution

(a) 5y−1

Notice the exponent applies to just the base y . 5y1
Take the reciprocal of y and change the sign of the exponent. 51y1
Simplify. 5y

(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. 1(5y)1
Simplify. 15y

(c) (−5y)−1

The base is −5y . Take the reciprocal of −5y and change the sign of the exponent. 1(5y)1
Simplify. 15y
Use ab=ab. 15y
Exercise 10.8.9:

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

Answer a

8p

Answer b

18p

Answer c

18p

Exercise 10.8.10:

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

Answer a

11q

Answer b

111q

Answer c

111q

Now that we have defined negative exponents, the Quotient Property of Exponents needs only one form, aman=amn, 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, an=1an.

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 am • an = am + n
Power Property (am)n = am • n
Product to a Power Property (ab)m = ambm
Quotient Property aman = am − n, a ≠ 0, m > n
  aman=1anm, a ≠ 0, n > m
Zero Exponent Property a0 = 1, a ≠ 0
Quotient to a Power Property (ab)m=ambm, b ≠ 0
Definition of a Negative Exponent an=1an
Example 10.8.6:

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

Solution

(a) x−4 • x6

Use the Product Property, am • an = am + n. x4+6
Simplify. x2

(b) y−6 • y4

The bases are the same, so add the exponents. y6+4
Simplify. y2
Use the definition of a negative exponent, an=1an. 1y2

(c) z−5 • z−3

The bases are the same, so add the exponents. z53
Simplify. z8
Use the definition of a negative exponent, an=1an. 1z8
Exercise 10.8.11:

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

Answer a

x4

Answer b

1y5

Answer c

1z9

Exercise 10.8.12:

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

Answer a

a5

Answer b

1b4

Answer c

1c15

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 10.8.7:

Simplify: (m4n−3)(m−5n−2).

Solution

Use the Commutative Property to get like bases together. m4m5n2n3
Add the exponents for each base. m1n5
Take reciprocals and change the signs of the exponents. 1m11n5
Simplify. 1mn5
Exercise 10.8.13:

Simplify: (p6q−2)(p−9q−1).

Answer

1p3q3

Exercise 10.8.14:

Simplify: (r5s−3)(r−7s−5).

Answer

1r2s8

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

Example 10.8.8:

Simplify: (2x−6y8)(−5x5 y−3).

Solution

Rewrite with the like bases together. 2(5)(x6x5)(y8y3)
Simplify. 10x1y5
Use the definition of a negative exponent, an=1an. 101x1y5
Simplify. 10y5x
Exercise 10.8.15:

Simplify: (3u−5v7)(−4u4v−2).

Answer

12v5u

Exercise 10.8.16:

Simplify: (−6c−6d4)(−5c−2d−1).

Answer

30d3c8

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

Example 10.8.9:

Simplify: (k3)−2.

Solution

Use the Product to a Power Property, (ab)m = ambm. k3(2)
Simplify. k6
Rewrite with a positive exponent. 1k6
Exercise 10.8.17:

Simplify: (x4)−1.

Answer

1x4

Exercise 10.8.18:

Simplify: (y2)−2.

Answer

1y4

Example 10.8.10:

Simplify: (5x−3)2.

Solution

Use the Product to a Power Property, (ab)m = ambm. 52(x3)2
Simplify 52 and multiply the exponents of x using the Power Property, (am)n = am • n. 25k6
Rewrite x−6 by using the definition of a negative exponent, an=1an. 251x6
Simplify. 25x6
Exercise 10.8.19:

Simplify: (8a−4)2.

Answer

64a8

Exercise 10.8.20:

Simplify: (2c−4)3.

Answer

8c12

To simplify a fraction, we use the Quotient Property.

Example 10.8.11:

Simplify: r5r4.

Solution

Use the Quotient Property, aman=amn. r5(4)
Be careful to subtract 5 - (\textcolor{red}{-4}).  
Simplify. r9
Exercise 10.8.21:

Simplify: x8x3.

Answer

x11

Exercise 10.8.22:

Simplify: y7y6.

Answer

y13

Contributors and Attributions


This page titled 10.8: Integer Exponents and Scientific Notation (Part 1) is shared under a not declared license and was authored, remixed, and/or curated by OpenStax.

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