Welcome to Unit 5Our Topics for this week

• Radical Exponents– Review Rules for Exponents

– Zero exponents

– Negative Exponents

• Rational Expressions

• Simplifying Radicals

• Operations with Radicals

Laws of Exponents

• PRODUCT RULE OF EXPONENTS

(ax) * (ay) = a(x + y)

(KEEP THE BASE and ADD THE EXPONENTS.)

EXAMPLES:Simplify: x3 * x2

Laws of Exponents

• QUOTIENT RULE OF EXPONENTS

(ax) / (ay) = a(x - y) (KEEP THE BASE and SUBTRACT THE EXPONENTS)

EXAMPLE:

57 / 55

= 57-5

= 52

Additional Laws of Exponents

• Anything to the zero power is 1. a0=1

• Anything to the first power is itself. a1=a

• A negative exponent moves the term to the other side of the fraction bar.

a-1 = 1/a and 1/a-1 = a

Eliminating Negative Exponents

• Move term to other side of fraction bar.a-1 = 1/a and 1/a-1 = a

EXAMPLE:

y3 / y14

= y3 – 14

= y-11

Remove the negative sign

= 1/y11

Laws of Exponents

• POWER RULE OF EXPONENTS (ax)y = axy (KEEP THE BASE and MULTIPLY THE EXPONENTS.)

EXAMPLE:

(m11)4

= m11*4

= m44

More Examples - Power Rule

When there are TWO factors

(ax3)2 = a1*2 * x2 * 3 = a2x6

EVALUATE:

(x/y)3 =

Example 33 2 5

4 1

3

4

x y z

ab c

=

=

=

Raise each factor to the –3 power

Move all factors with negative exponents such that the exponents are positive

Calculate 43 and 33, and alphabetize the variables

Example 33 2 5

4 1

3

4

x y z

ab c

3 9 6 15

3 3 12 3

3

4

x y z

a b c

3 9 15 3

3 12 3 6

4

3

x z a

b c y

3 9 15

12 3 6

64

27

a x z

b c y

=

=

=

Raise each term to the –3 power

Move all terms with negative exponents such that the exponents are positive

Calculate 43 and 33, and alphabetize the variables

Examples of Rational Exponents

Rational exponents are exponents that can be expressed in the form of a fraction.

x1/2

b2/3

c40

Adding – with Rational Exponents

x1/2 + 5x1/2 (Like Terms, add)

= 6x1/2

4y1/2 + 6y1/3 (Powers of y are not

the same, so we

cannot add. Done.)

Multiplying with exponents (RULE: Add exponents)

(3x1/2)(4x1/3)

= 12x1/2+1/3 Multiply coefficients

= 12x3/6+2/6 Add exponents, LCD

= 12x5/6

First divide coefficients, then subtract exponents

= (-40/5) a9/8 – ¼ b2 – 1/3 c1 – 1

Find LCD for exponents

= -8 a9/8 - 2/8 b6/3-1/3 c0

Leave exponents as improper fractions

= -8 a7/8 b5/3

9/8 2

1/ 4 1/3

40

5

a b c

a b c

Dividing with exponents

Example: (x1/2)1/3

= x(1/2)(1/3)

= x1/6

Example: (a4/5b2/3)1/7

= a(4/5)(1/7)b(2/3)(1/7)= a4/35b2/21

“Power to a Power” Example

Simplify: (16x3y4z3/8)1/2

First, multiply exponents

= 16(1)(1/2)x(3)(1/2)y(4)(1/2)z(3/8)(1/2)

= 161/2x3/2y4/2z3/16

Now convert number to radical form, reduce exponents

= √16 x3/2y2z3/16

Extract the square root

= 4x3/2y2z3/16

Example:

Radicals

• Radicals are roots. The typical radical symbol √ is considered to be a “square root” symbol. In WORD, use Insert, Symbol.

• √ [ 4] would be square root of 4.

• In TEXT, such as in the discussion, we write SQRT, for example √ [3]= SQRT[3].

Examples of Radicals

√ [8] is “the square root of eight”The index is an understood 2 and the radicand is 8.

______√100a2b is “the square root of one hundred a squared b”

The index is an understood 2 and the radicand is 100a2b. ____ 3√27c6 is “the cube root of twenty-seven c to the sixth power”

The index is 3 and the radicand is 27c6. ___5√-32 is “the fifth root of negative thirty-two”

The index is 5 and the radicand is –32.

Terms with rational exponents are related to terms with radicals. Here’s how.

___am/n = n√am

When there is a fractional exponent, the numerator is a power, denominator is the index of the radical.

Example: ___x2/3 = 3√x2

More Examples:

___

x2/3 = 3√x2

____

2004/7 = 7√2004

________ _____

(36a2b4)1/2 = 2√(36a2b4)1 = √36a2b4

Simplifying Radicals

√[25] is “the square root of twenty five”

The index is an understood 2 and the radicand is 25.

The simplified answer is +5 or -5

√[100] = 10 because 10*10 = 100

√[49] = 7 because 7*7 = 49

Simplifying Radicals

Perfect squares:

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, …

A constant or variable with an EVEN exponent is also a perfect square:

x2, x4, x6, x8, x10, x12, x14, …

Simplifying Radicals

Perfect cubes:

1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, …

A constant or variable with an exponent that is a MULTIPLE OF 3 is also a perfect cube:

x3, x6, x9, x12, x15, x18, x21, …

Example: Evaluate 163/2

Keep in mind that this can be rewritten like this: ___ ___2√163, or just √163

(The denominator of the fractional exponents gives the root, and the numerator is a power)

Because radicals and exponents are considered to be the “same level” in the order of operations, you can either deal with the radical first and the exponent second, or the exponent first and the radical second.

Evaluating Fractional Exponents

You can either deal with the radical first and the exponent second, or the exponent first and the radical second.

We will look at both ways.

ROOT first: ___√163 = 43 = 64The square root of 16 is 4, and 43 is 4*4*4 = 64

RAISING to power first: ___ ____√163 = √4096 = 6416 cubed is 16*16*16 = 4096, and the square root of 4096 is 64.

Evaluating Fractional Exponents

Example: Evaluate 45/2

Root first __√45 = 25 = 32

The square root of 4 is 2, and 25 is 2*2*2*2*2 = 32

RAISING to power first: ___ ____√45 = √1024 = 324 to the fifth power is 4*4*4*4*4 = 1024, and the square root of 1024 is 32.

EXAMPLE:

Here are examples of when you might want to convert from radical form to rational exponent form:

___√x30 = x30/2 = x15

__3√y27 = y27/3 = y9

Changing Radical to Exponent Form

Simplify: ___4√x28

Example:

Simplify: ___4√x28

= x28/4

= x7

Example:

Practice Problems

Example – Multiply, given Fractional Exponents

EVALUATE:

(2x½)(3x⅓)

=

Example – DISTRIBUTE, given Fractional Exponents

EVALUATE:

-2x5/6(3x1/2 – 4x-1/3 )

=

MORE PRACTICE