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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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Section P.2

Exponents and Radicals


Definition of Exponent

  • An exponent is the power p in an expression ap.

    52

  • 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

(-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 – 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 – 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 exponents… multiply the exponents.


Definition of nth root of a number

Let a and b be real numbers and let n ≥ 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.


Cube roots are a different story.

Must have 3 of a kind


Cube Roots

Must have 3 of a kind


More Cube Roots


Simplify first

3

2


Simplify, rationalize


Reduce and rationalize


Cube root


P.2 Assignment

  • Page 21

  • #9 – 42 multiples of 3 (a’s only),

  • 55 – 63 odd (a’s only)

  • 72 – 84 Multiples of 3 (a’s only)

  • 95 – 99 odd (a’s only)


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