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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

Section P.2

Exponents and Radicals


Definition of exponent

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

Puzzler

(-5)2 = -52?

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

25 = -25


Examples

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

Multiplication with Exponents by Definition

3235

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

= 37

Note 2+5=7


Property 1 for exponents

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?


Which one is it1

Which one is it?


Examples of property 1

Examples of Property 1


By the definition of exponents

By the Definition of Exponents

Notice that

5 – 3 = 2


Examples1

Examples


Property 2 for exponents

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

Negative Exponents

Notice that

3 – 5 = -2


Property 3 definition of negative exponents

Property 3Definition of Negative Exponents

  • If nis a positive integer, then


Examples of negative exponents

Examples of Negative Exponents

Notice that: Negative Exponents do not indicate negative numbers.

Negative exponents do indicate Reciprocals.


Examples of negative exponents1

Examples of Negative Exponents

Notice that exponent does not touch the 3.


Zero as an exponent

Zero as an Exponent


Zero to the zero

Zero to the Zero?

Undefined

STOP

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


Examples2

Examples


Property 5 for exponents

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

Examples of Property 5


Power to a power by definition

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

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 5 + 6


Examples of property 6

Examples of Property 6

One base, two exponents… multiply the exponents.


Definition of n th root of a number

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

Property 2 for Radicals

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


Property 3 for radicals

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

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

10

4

8

3

3

2


Example 1

Example 1


Example 2

Example 2


Example 3

Example 3


Simplified form for radical expressions

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 6


Example 7

Example 7


Example 10

Example 10


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

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


Rationalize the denominator

Rationalize the denominator.

This will always be a perfect square.


Rationalize the denominator1

Rationalize the denominator.

Often you will not need to write this step.


Rationalize the denominator2

Rationalize the denominator.

5


Simplify numerator first if possible

Simplify numerator first, if possible


Simplify first then rationalize

Simplify first then rationalize.


Reduce simplify rationalize

Reduce, simplify, rationalize


Product

Product


Another product

Another product


Assume that r and t represent nonnegative real numbers

Assume that r and t represent nonnegative real numbers.


Cube roots are a different story

Cube roots are a different story.

Must have 3 of a kind


Cube roots

Cube Roots

Must have 3 of a kind


More cube roots

More Cube Roots


Simplify first

Simplify first

3

2


Simplify rationalize

Simplify, rationalize


Reduce and rationalize

Reduce and rationalize


Cube root

Cube root


P 2 assignment

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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