11-1 Simplifying Radicals

In the expression , is the radical sign and

64 is the radicand.

If x2 = y then x is a square root of y.

1. Find the square root:

8

2. Find the square root:

-0.2

64

64

0.04

11, -11

4. Find the square root:

21

5. Find the square root:

3. Find the square root: 121

441

25

815

9

6.82, -6.82

6. Use a calculator to find each square root. Round the decimal answer to the nearest hundredth.

46.5

1 • 1 = 12 • 2 = 43 • 3 = 9

4 • 4 = 165 • 5 = 256 • 6 = 36

49, 64, 81, 100, 121, 144, ...

What numbers are perfect squares?

Multiplication property of square roots

Properties of Radicals

Division property of square roots

Properties of Radicals

What does this really mean?

it can be rewritten as:

which we all know equals 10

or

which we all know equals 10

How can I use this?To write a radical in simplest form you must make sure:•The radicand has no perfect square

factors•The radicand has no fractions

•The denominator of a fraction has no radicalThis property addresses the first point

To simplify

Find a perfect square that goes into 75.

2. Simplify

Find a perfect square that goes into 600.

Simplify

1. .

2. .

3. .

4. .

2 18

72

3 8

6 236 2

Look at these examples and try to find the pattern…

How do you simplify variables in the radical?

x7

1x x2x x

4 2x x

6 3x x

What is the answer to ? x7

7 3x x x

As a general rule, divide the exponent by two. The

remainder stays in the radical.

Find a perfect square that goes into 49.

4. Simplify 49x2

5. Simplify 258x

122 2x x

Simplify 369x

1. 3x6

2. 3x18

3. 9x6

4. 9x18

Multiply the radicals.

6. Simplify 6 10

60

2 15

7. Simplify Multiply the coefficients and radicals.

Simplify

1. .

2. .

3. .

4. .

24 3x44 3x

2 48x448x

How do you know when a radical problem is done?

1. No radicals can be simplified.Example: not done because 4 is a factor

2. There are no fractions in the radical.Example: not done because it is a fraction

3. There are no radicals in the denominator.Example: not done because radical 5 is in

denominator

8

1

4

1

5Division property of square

roots helps with points 2 and 3

8. Simplify.

Divide the radicals.

108

3

108

3

366

Uh oh…There is a

radical in the denominator!

Whew! It simplified!

9. Simplify

8 2

2 8

2

Uh oh…Another

radical in the denominator!

Whew! It simplified again! I hope they all are like this!

10. Simplify

5

7

5

7

35

49 35

7

Since the fraction doesn’t reduce, split the radical up.

Uh oh…There is a fraction in the radical!

How do I get rid of the radical in

the denominator?

Multiply by the “fancy one” to make the denominator a

perfect square!