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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.

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)

ExamplesWhich of these will be negative?

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.

Property 2 for Exponents

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

To divide like bases subtract the exponents.

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.

Property 5 for Exponents

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

Distribute the exponent.

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

Example 1 match the index.

Example 2 match the index.

Example 3 match the index.

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 match the index.

Example 7 match the index.

Example 10 match the index.

Rationalize the denominator. the index.

This will always be a perfect square.

Rationalize the denominator. the index.

Often you will not need to write this step.

Rationalize the denominator. the index.

5

Simplify numerator first, if possible the index.

Simplify first then rationalize. the index.

Reduce, simplify, rationalize the index.

Product the index.

Another product the index.

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

Cube roots are a different story. the index.

Must have 3 of a kind

Cube Roots the index.

Must have 3 of a kind

More Cube Roots the index.

Simplify, rationalize the index.

Reduce and rationalize the index.

Cube root the index.

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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