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# 7.1 Properties of Exponents - PowerPoint PPT Presentation

7.1 Properties of Exponents. ©2001 by R. Villar All Rights Reserved. Properties of Exponents. Consider the following… If x 3 means x • x • x and x 4 means x • x • x • x then what is x 3 • x 4 ? x • x • x • x • x • x • x x 7

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7.1 Properties of Exponents

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## 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?

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

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

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