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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

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.





By the definition of exponents
By the Definition of Exponents

Notice that

5 – 3 = 2



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 to the zero
Zero to the Zero?

Undefined

STOP

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



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.



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 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



Example 1
Example 1 match the index.


Example 2
Example 2 match the index.


Example 3
Example 3 match the index.


Simplified form for radical expressions
Simplified Form for Radical Expressions match the index.

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 match the index.


Example 7
Example 7 match the index.


Example 10
Example 10 match the index.



Rationalize the denominator
Rationalize the denominator. the index.

This will always be a perfect square.


Rationalize the denominator1
Rationalize the denominator. the index.

Often you will not need to write this step.






Product
Product the index.


Another product
Another product the index.


Assume that r and t represent nonnegative real numbers
Assume that the index.r and t represent nonnegative real numbers.


Cube roots are a different story
Cube roots are a different story. the index.

Must have 3 of a kind


Cube roots
Cube Roots the index.

Must have 3 of a kind


More cube roots
More Cube Roots the index.


Simplify first
Simplify first the index.

3

2




Cube root
Cube root the index.


P 2 assignment
P.2 Assignment the index.

  • 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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