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7.1 Properties of Exponents

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7.1 Properties of Exponents

©2001 by R. Villar

All Rights Reserved

Properties of Exponents

Consider the following…

If x3meansx • x • x

and x4meansx • x • x • x

then what isx3 • x4 ?

x • x • x• x • x • x • x

x7

Can you think of a quick way to come up with the solution?

Add the exponents.

Your short cut is called the Product of Powers Property.

Product of Powers Property:

For all positive integers m andn:am •an = am + n

Example: Simplify (3x2y2)(4x4y3)

Mentally rearrange the problem (using the commutative and associative properties).

(3 • 4) • (x2 • x4) •(y2• y3)

12x6y5

How do you raise a power to another power?Example: Simplify (x2)3

This means x2 • x2 • x2

Using the Product of Powers Property gives x6

What is the short-cut for getting from (x2)3 to x6 ?

Multiply the exponents.

This short-cut is called the Power of a Power Property.

Power of a Power Property:

For all positive integers m andn:(am) n = am • n

Example: Simplify (2m3n5)4

Raise each factor to the 4th power.

(24) • (m3)4 • (n5)4

16m12n20

The last problem was an example of how to use the Power of a Product Property.

Power of a Product Property:

For all positive integers m:(a • b)m = am • bm

Division Properties of Exponents

a • a • a • a • a

a • a • a

Notice that you

can cancel from

numerator to

denominator.

Let’s look at

each problem in

factored form.

= a2

How do you divide expressions with exponents?

Examples:a5 = a3x3 =x5

x • x • x

x • x • x • x • x

= 1

x2

Do you see a “short-cut” for dividing these expressions?

The short-cut is called the Quotient of Powers Property

Quotient of Powers Property:

am = am – nan

a ≠ 0

This means that when dividing with the same base, simply subtract the exponents.

Examples:a5 = a3x3 =x5

= a2

a5 – 3

x3 – 5

= x–2

= 1

x2

One final property is called the Quotient of Powers Property. This allows you to simplify expressions that are fractions with exponents.

Quotient of Powers Property:

Example: Evaluate

This is the same as 33 43

= 27

64

Here’s a tool you may want

to use to help you remember

the properties for exponents.

Stairway to the Exponents

Power

The steps represent the Order

of Operations. When working

with exponents, step down to

the next lower step.

Mult/Divide

Add/Subtract

mult/divadd/subtmove down a step

For example, when multiplying

expressions with exponents, step down and add the exponents.

Negative & Zero Exponents

1

3

1

9

1

27

243

81

27

9

3

1

What do you think 3–4 will be?

3–4 = 1 = 1

3481

Study the table and think about the pattern.

Exponent, n 5 4 3 2 1 0 –1 –2 –3Power, 3n

This pattern suggests two definitions:

Negative Exponents:

a–n = 1 an

a cannot be zero

Zero Exponents:

a0 = 1

a cannot be zero

Example: Simplify 3y –3 x–2

= 3x2 ` y3

Example: Simplify 3–8• 35

Step down and add the exponents

3–3

=1

33

= 1

27

This is the same as 3 • 1 • x2 1 y31

Example: Simplify (2x4)–2

Step down and multiply the exponents

2–2 •x–8

= 1

4x8

= 1 • 1

22 x8

Remember, anything (other than 0) raised to the zero power is equal to 1 by definition.

Example: (–8)0

= 1

Example 5(–200x–6y –2 z 20)0

= 5(1)

= 5