Simplifying Radicals

Unit VIII, Lesson 4Online AlgebraVHS@PWCS

Square Root ReviewFind the principal square root of each of the following.

1.

2.

3.

4

1. 9

2. 5

3. About 11.8

4. About 14.1

8125

125

200

Square Root Review

Recall from lesson VIII.3 that a square root is defined as:

If a2 = b, then a is a square root of b.

So far we have used the decimal form of any square root that is not a perfect square (a number that as a whole number for a square root). Now we will leave our square roots in radical form.

Radical Form

When you are asked to leave an answer in radical form, it means to leave any square roots in your answer.

Examples:

4 5 10 1513 2

Radical Form

There are some rules we have to following when simplifying radicals. A radical is in simplest form if the following is true:

1. All numbers under the square root symbol have no perfect square factors.

2. The expression under the radical does not contain fractions.

3. The denominator does not contain a radical expression.

What does all this mean. Click to find out!

Simplifying radicals

Simplify

Our first rule says that we can not have numbers with a perfect square factor under the radical sign.

125 has a factor of 25. Factors are numbers that divide the product without a remainder.

25 x 5 = 125

We can rewrite as Since , we can write as

125

125 25 525 5 125 5 5

Simplifying Radicals

That probably sounded difficult, but it is pretty easy when broken down. To find the square root of 45.

1. Find all the prime factors of 45. I use a factor tree.2. Rewrite using the prime factors

3. Pull out pairs of factors. In this case the 3’s.

Though we have pulled out pairs we only use one 3 in our answer, because 3(3) is 9 and the square root of 9 is 3.

The only factors of 5 are 1 and 5 so the only perfect square under the square root symbol is 1 and our square root

is simplified.

45

9 5

3 3

45 3 3 5

3 5

Simplifying Radicals

Find all the factors of 240? Rewrite using the prime factors

Pull out pairs.

Multiply numbers outside the radical sign and then those under the radical sign.

240

24 10

2 12 5 2

3 4

2 2

240 2 2 2 2 3 5

2 22 2 3 5 2 2 3 5

4 15

Try these on your own. Click for the answers.1.

2.

3.

4.

50

300

28

80

5 2

10 3

2 7

4 5

1.

2.

3.

4.

Remember our first rule: No

perfect square factors under

the radical sign!

Multiplying square roots.Multiplication property of square roots

For all a and b greater than or equal to 0: So to multiply 1. Find all the factors of 8 and 6.

2. Write under one radical.

3. Pull out all pairs.

4. Multiply

8

4 2

2 2

6

2 3

8 6

2 2 2 2 3

2 2 2 2 3

2 2 3

4 3

Multiplying Square Roots – Try these!

30 6

12 27

14 8

15 35

1.

2.

3.

4.

6 518

4 7

5 21

1.

2.

3.

4.

Dividing Square Roots

Our last 2 rules deal with fractions or dividing square roots.

The expression under the radical does not contain fractions.

The denominator does not contain a radical expression.

Dividing Square Roots

For any numbers a > 0 and b > 0:

So to simplify

1. Split in to 2 square roots

2. Take the square root of each and simplify.

a ab b

2564

2564

58

Simplify:

1. Find the each square root. 2. Simplify if needed.

969

4 63

2 2 2 2 2 33 3

2 2 2 33

Review – Are the following radical expressions in simplest form?

1.

2.

3.

4.

1. No 12 has 4 as a factor which is a perfect square.

2. Yes the factors of 30 are 2, 3, and 5. None of which are perfect squares.

3. No there is a radical in the denominator. The square root of 4 is 2. So our answer is 5/2

4. No the 2’s can be canceled so our answer is

2 12

5 30

54

2 32 3