Rationalizing Radical Expressions

Section 7.5

What does it mean to “rationalize the denominator?”

2

5

Recall: A rational number is a terminating or repeating decimal.

How can we change the denominator into a whole

(rational) number?

We prefer to have whole numbers in the denominators.

It makes them easier to combine.

Rationalize the denominator:

2

5

Relax! It is easy to change an irrational into a rational!

2

52 5

5If possible, simplify the

denominator and/or numerator. Multiply the denominator by the

same irrational number. Squaring a square root cancels it out.

Multiply the numerator as well so the value doesn’t change.

5 5 25 5

5

5

Rationalize the denominator:

2 9

16y

Relax! It is easy to change an irrational into a rational!

y

y 3

2

y

y

If possible, simplify the denominator and/or numerator. Multiply the denominator by the part that is

irrational. Multiply the numerator as well so the value doesn’t change.

2 3

4 y

6

4 y

Rationalize the denominator:

32

25

Relax! It is easy to change an irrational into a rational!

3

3

2

25

3 10

5

If possible, simplify the denominator and/or numerator. When working with roots other than square roots, multiply the denominator by the

remaining amount needed to cancel the denominator and make it

rational. Multiply the numerator as well so the value doesn’t change. 3 5 5 5 5

We need another 3 5

3

3

5

5

Rationalize the denominator:5 2

5 1232

a

bIf possible, simplify the

denominator and/or numerator. When working with roots other than square roots, multiply the denominator by the remaining amount needed to cancel the

denominator and make it rational. Multiply the numerator

as well so the value doesn’t change.

5 5b bWe need another 5 3b

5 2

2 5 22

a

b b

5 2 3

32

a b

b

5 3

5 3

b

b

What does it mean to “rationalize the denominator?”

3

2 5 1

Recall: A rational number is a terminating or repeating decimal.

How can we change a denominator like this into

a whole (rational) number?

We prefer to have whole numbers in the denominators.

It makes them easier to combine.

Properties and Rules for Radicals

Product Rule for Radicals

Quotient Rule for Radicals

Like radicals

Conjugates

n na b

na

b

Radicals with the same radicand and index/root. We can only add/subtract like radicals.

0b

n ab

n

n

a

b

The conjugate of (a + b) is (a – b). It follows that (a + b) (a – b) = a2 – b2

3 5;3 5

1; 1x x

3 5; 3 5

Rationalize the denominator:

3

2 5 1When working denominators of

sums or differences, multiply the denominator by the conjugate to

make it rational. Multiply the numerator as well so the value

doesn’t change.

(2 5 1)(2 5 1) 4 5 1

2 5 1

3(2 5 1)

19

The conjugate is

3 (2 5 1)

2 5 1 (2 5 1)

6 5 3

19

Rationalize the denominator:

5 3

2x

When working denominators of sums or differences, multiply the denominator by the conjugate to

make it rational. Multiply the numerator as well so the value

doesn’t change.

2( 2)( 2) 2x x x

2x The conjugate is

5 3 ( 2)

2 ( 2)

x

x x

2

( 5 3)( 2)

2

x

x

2

3 5 10 3 2

2

x x

x

Can we also “rationalize the numerator?”

5

2

Yes!

The process is the same. The objective is

to make the numerator rational,

just as we did for the denominator of the previous examples.

5

2 5

Solving Radical Equations

Section 7.6

Solving

Radical Equations

One radical in the problem

Two radicals in the problem

Only two radicals in the problem and

nothing else

Two radicals in the problem and other stuff not under a

radical

3

4

and

and

x and

and x

x and

and

Opposite Operations

The Big Picture.

There are 3 different types of problems.

2x

nxn x

3 x

4x

Solving Radical Equations

One radical in the problem Get radical alone on one side of = Apply same power as root (opposite

operation) to both sides to cancel the radical

Solve for x and check solution

3 2 5x

3 5 2 1x

9 3x x

Solving Radical Equations

Two radicals in the problem

Only two radicals in the problem and nothing else

Get each radical alone on one side of =

Apply same power as root (opposite operations) to both sides to cancel the radical

Solve for x and check solution

3 36 1 2 5x x

2 1 1 2x x

Solving Radical Equations

Two radicals in the problem

Two radicals in the problem and other stuff not under a radical

Get one radical alone to one side of =Get other radical and extra stuff alone on other side of =Apply same power as root (opposite operation) to both sides to cancel one radicalGet remaining radical alone on one side of = and apply same power as root to both sides and cancel the radicalSolve for x and check solution

3 1 3 2x x

Application: Finding the missing side of a right triangle

9 units?

6 units

How do find the missing side

length?

2 2 2leg leg hyp 2 2 2a b c 2 2 26 9a 2 81 36a

2 45a The opposite of a

square is a square root

b=

Hypotenuse

c = a = 45 3 5a