Section P.2. Exponents and Radicals. Definition of Exponent. An exponent is the power p in an expression a p . 5 2 The number 5 is the base . The number 2 is the exponent . The exponent is an instruction that tells us how many times to use the base in a multiplication. Puzzler.

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Section P.2

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The exponent is an instruction that tells us how many times to use the base in a multiplication.

Puzzler

(-5)2 = -52?

(-5)(-5) = -(5)(5)

25 = -25

43

-34

(-2)5

(-3/4)2

=(4)(4)(4) = 64

=(-)(3)(3)(3)(3) = -81

=(-2)(-2)(-2)(-2)(-2)= -32

=(-3/4)(-3/4) = (9/16)

Examples

Which of these will be negative?

Multiplication with Exponents by Definition

3235

= (3)(3)(3)(3)(3)(3)(3)

= 37

Note 2+5=7

Property 1 for Exponents

If a is any real number and r and s are integers, then

To multiply two expressions with the same base, add exponents and use the common base.

Which one is it?

Which one is it?

Examples of Property 1

By the Definition of Exponents

Notice that

5 â€“ 3 = 2

Examples

Property 2 for Exponents

If a is any real number and r and s are integers, then

To divide like bases subtract the exponents.

Negative Exponents

Notice that

3 â€“ 5 = -2

Property 3Definition of Negative Exponents

If nis a positive integer, then

Examples of Negative Exponents

Notice that: Negative Exponents do not indicate negative numbers.

Negative exponents do indicate Reciprocals.

Examples of Negative Exponents

Notice that exponent does not touch the 3.

Zero as an Exponent

Zero to the Zero?

Undefined

STOP

Zeros are not allowed in the denominator. So 00 is undefined.

Examples

Property 5 for Exponents

If a and b are any real number and r is an integer, then

Distribute the exponent.

Examples of Property 5

Power to a Power by Definition

(32)3

= ((3)(3))1((3)(3))1((3)(3))1

= 36

Note 3(2)=6

Property 6 for Exponents

If a is any real number and r and s are integers, then

A power raised to another power is the base raised to the product of the powers.

Examples of Property 5 + 6

Examples of Property 6

One base, two exponentsâ€¦ multiply the exponents.

Definition of nth root of a number

Let a and b be real numbers and let n â‰¥ 2 be a positive integer. If

a = bn

then b is the nth root of a.

If n = 2, the root is a square root.

If n = 3, the root is a cube root.

Property 2 for Radicals

The nth root of a product is the product of nth roots

Property 3 for Radicals

The nth root of a quotient is the quotient of the nth roots

For a radicand to come out of a radical the exponent must match the index.

10

4

8

3

3

2

Example 1

Example 2

Example 3

Simplified Form for Radical Expressions

A radical expression is in simplified form if

1. All possible factors have been removed from the radical. None of the factors of the radicand can be written in powers greater than or equal to the index.

2. There are no radicals in the denominator.

3. The index of the radical is reduced.

Example 6

Example 7

Example 10

For something to go inside a radical the exponent must match the index.

Rationalize the denominator.

This will always be a perfect square.

Rationalize the denominator.

Often you will not need to write this step.

Rationalize the denominator.

5

Simplify numerator first, if possible

Simplify first then rationalize.

Reduce, simplify, rationalize

Product

Another product

Assume that r and t represent nonnegative real numbers.