Radical EquationsAn equation in which the variable appears under a radical sign is called a radical equation.2 3 4 7x

Isolate the radical2 3 3x

Square both sides 2 2

2 3 3x

2 3 9x

Solve the equation2 12x

6x

Check your answer2 36) 4( 7

9 4 7

3 4 7

4 8x x

4 8x x

2 2

4 8x x

24 16 64x x x 2 17 60 0x x ( )5 2 0( 1 )x x

5 12x x ( ) 85 54

9 8 5

3 8 5

( )12 124 8

16 8 12

4 8 12 Reje

ct

Solving Radical Equations Graphically

2 3 4 7x

Enter each equation in the Y= window.1

2

2 3 4

7

Y

Y

x

Press GRAPH

Press 2nd CALC 5 (intersect)

Move the curser until it is on the curve.

Press ENTER ENTER ENTER

The calculator calculates the point of intersection.

6

7

x

y

4 8x x

1

2

4 8x

x

Y

Y

5

5

x

y

Fractional Equations

A fractional equation is an equation in which the variable appears in the denominator of one or more fractions.

To solve a fractional equation, eliminate the denominators by multiplying everything by the LCD of all the denominators.

2 9 110 5x x

LCD = 10x

Multiply by the LCD

10 1( ) (2

0 10) ( )9 110 5

x xx x

x

Cancel all denominators( ) ( )10 )2 (29 1x

Solve the equation20 9 2x 9 18x

2x

2{ }x

1 1 133 8x x x

LCD = (8x)(x +

3)

8 3) 8 3) 81 1 13

3( )( ( )( ( )( 3)

8x x x x

x x xx x

( )( ) ( )18 1 1 )3 8 3(3x x x

8 24 8 13 39x x x 3 15x

5x

5{ }x

More Fractional Equations

2

1 82 2 4

xx x x

Factor in order to find the LCD.

1 8(2 2 ) )2 2(

xx x x x

LCD = (x + 2)(x -

2)

2 2 2 2( )( ) ( )( ) ( )( )(

1 82 2 2

2)( )

22

xx x x x

x x x x x x

2 1( ) ( 8)2x x x 2 3 10 0x x

5)2 0( )(x x 2 5x x

Don’t forget to check for

extraneous roots.

If x = –2, then one of the fractions would be undefined. Therefore, -2 is an extraneous root.

Reject 5{ }x

That was easy

Radical vs Exponential Form

All radicals can be written in exponential form.

n ax na

x

All exponents can be written in radical form.

na

x n ax

3 4x

x43

52

x x 5 2

Equations with Rational Exponents

That doesn’t sound easy.

Don’t worry. You’ll be pushing the easy

button in just a few minutes.

An equation with a rational exponent is an equation in which the variable is raised to an exponent that is a fraction.

To solve an equation with a rational exponent, raise both sides to the reciprocal power of the variable in order to transform the exponent of the variable to 1 .

Solving Equations with Rational Exponents

32 9 1x

Isolate the variable

32 8x

Raise both sides to the reciprocal of the exponent of the variable.

3

32

223

8x

21 3 8x Simplify the equation.

3 64x

4x

54 7 9x

54 2x

5

54

445

2x

41 5 2x

5 16x

1.7411x

Rationalizing DenominatorsIn order for a fraction to have a rational denominator, there cannot be a radical in the denominator. The process of rewriting the fraction in an equivalent form without a radical in the denominator is called Rationalizing the Denominator.

3

5Since there is a radical in the denominator, we must rationalize the denominator.

3

55

5

Multiplying the radical by itself will remove it from the denominator.

3 55

The new fraction is an equivalent form of the original fraction.

They are Sam Ting!Let’s try one more.

4

3 22

2

24 23

4 26

2 2

3

Multiplying by the Conjugate Pair

Conjugate Pairs are two binomial expressions that are the same with the exception of the sign in the middle.

(4 )7 and (4 )7 are Conjugate Pairs.

To rationalize the denominator of a fraction that has a binomial radical expression, you must multiply by the Conjugate Pair.

8

3 73 7

3 7

24 8 7

9 7

24 8 72

12 4 7

5

2 6 32 6 3

2 6 3

10 30 34 3 )3( 6

10 30 3

4 108

10 30 3104

5 15 3

52

Radical Equation Word Problem

The number of seconds in the period of a pendulum is the length of time required for the pendulum to make one complete swing back and forth.

232L

T The formula gives the period, T, for a pendulum of length L infeet. If Todd wants to build a grandfather clock with a pendulum that swings back and forth every 2 seconds, how long, to the nearest tenth of a foot, should he make the pendulum?

L is the variable and T = 2.

2 232L

22 32

L

132L

22

321 L

2

132L

2 32L

2

32L

3.242277877L

3.2 f eetL