Chapter 0 Section 6

Rational Exponents and Radicals

Radicals and Radical Notation

n bThe n is called the “index”—it indicates what root to find.

The b is called the radicand– it indicates the number you are trying to find the root of…

With radical notation, if n=2 we normally don’t put the 2 in for the index, we just know it as the “square root”. If n=3 we call it the “cube root”…but we must put the 3 in for the index.

Evaluating Square Roots

121

64

25

Evaluating Cube Roots

3 64

3 125

3 8

3 27

Summary of evaluating

even #

odd #

NOT a real number

IS a real number

Mixed Examples

49 5 32 81

Radicals as Fractional Exponents

There is a relationship between radical notation and exponents:

nn bb1

The index becomes a fractional exponent instead.

All of the properties of radicals work for fractional exponents too.

The denominator of a fractional exponent is the index.

Convert to radical form and evaluate

31

64 32

125 23

363 64

4

23 125

25

25

336

36

216

Properties of Radicals

nnn baab

n

n

n

b

a

b

a

Product property: The root of a product is the product of the roots

Quotient property: The root of a quotient is the quotient of the roots

bnn b aa )( Power property: The root of a power is the power of the root

Simplify the square roots

50 42108 yx 52340 cba

225

25

42336 yx

36 2xy

ccaba 422104

acabc 102 2

Simplifying algebraic expressions

64y63 643 y63= = 4y2

m4

n8

4=

m44

n84=mn2

*You want the exponents of the variables to be a multiple of the index.

More simplifying algebraic expressions

If the exponents on the variables are not multiples of the index, rewrite as a product so one of the exponents is a multiple of the index.

4 10844 cba 4 28844 ccba

4 24 884 4ccba

4 222 4ccab

Simplify the expressionsSimplest Radical Form…

1353 = 273 5

= 273 53

533=

xx 612 272x

2236 x2236 x

26x

6. 274 34

Evaluate the expressions

123 183 12 183= 2163= = 6

804

54

805

4= = 164 = 2

3SOLUTION SOLUTION

7.

232503

5

Numeric expressions with +/-Combining Like Radicals…just like with combining like variables, you may combine radical expressions if the index and radicand match.

233 23–=543 – 23 = 23273 23– 23(3 – 1)= = 2 23

If the index and radicands don’t match after you have simplified, you may not combine the radicals and radicands… they aren’t like.

139133 1393 1312

Add or Subtract

7157873

Rationalizing the Denominator

7

1

If a reduced fraction still has a radical in the denominator it is not really reduced…there is still some work to do…you need to “rationalize the denominator”, that means make the denominator a Rational number instead of an Irrational number.

Think: what times the square root of 7 gives 7…

7

1

5

1

5

1

5

5

5

1

5

5

103

2

10

10

103

2

103

102

15

10

Conjugates

If the denominator of the fraction also has an addition or subtraction along with the radical you need to use something called the “conjugate”…some examples of conjugate pairs are listed below.

115,115

72,72

53,53

Multiply each pair together (use the same rules for FOIL) and see if you notice something special…

When multiplying conjugates, the O and I steps can be skipped (they cancel each other out)

Simplify

62

2

Find the conjugate and multiply numerator and denominator by that conjugate.

62

62

62

2

2

624

62

Simplify

23

62

23

23

29

1263226

7

3263226

Properties

nm

a mna1

mn a

n ma

naa n

1

11

nma

a nm 1