Simplifying Square Roots and Imaginary Numbers

Simplifying Square Roots

By Susan Joyce

Method 1

1. Find the largest perfect square that is a factor of the number

2. Write the factor pair under the square root (radical) sign

3. Write as the product of two separate square roots

4. Take the square root of the perfect square. Keep the other number under the radical.​

​

​Simplify Square Roots

Multiple Choice

What do you need to find to simplify square roots

• perfect square factor

• square roots

• even factors

• factors of ten

Multiple Choice

What is the largest perfect square that goes into $$\sqrt{96}$$  ?

• 4

• 9

• 6

• 16

Multiple Choice

Simplify $$\sqrt{45}$$  

What is the largest perfect square that is a factor of 45? 9

Write your number as the product of the perfect square and the other number.

$$\sqrt[]{9\times5}$$  

Simplify.

• $$9\sqrt{5}$$  

• $$5\sqrt{9}$$  

• $$3\sqrt{5}$$  

• $$5\sqrt{3}$$  

Multiple Choice

 Simplify $$\sqrt{84}$$  

What is the largest perfect square that is a factor of 84? 4

Write your number as the product of the perfect square and the other number.

$$\sqrt[]{4\ x\ 21}$$  

Simplify.

• $$4\sqrt{21}$$  

• $$2\sqrt{21}$$  

• $$21\sqrt{2}$$  

• $$21\sqrt{4}$$  

Multiple Choice

Simplify $$\sqrt{150}$$  

What is the largest perfect square that is a factor of 150? 25

Write 150 as a product of the perfect square and its other factor.

$$\sqrt[]{25\times6}$$  

Simplify.

• $$5\sqrt{6}$$  

• $$6\sqrt{5}$$  

• $$10\sqrt{15}$$  

• $$3\sqrt{75}$$  

Multiple Select

Select all perfect square factors

• 9

• 3

• 6

• 64

• 25

Multiple Select

Select all perfect square factors

• 5

• 100

• 144

• 200

• 121

Multiple Choice

Simplify $$\sqrt{242}$$  

What is the largest perfect square that is a factor of 242? 121

Write 242 as factor of the perfect square and the other factor.

$$\sqrt[]{121\times2}$$  

Simplify.

• $$2\sqrt{11}$$  

• $$11\sqrt{2}$$  

• $$4\sqrt{23}$$  

• $$2\sqrt{11}$$  

Multiple Choice

Simplify $$\sqrt{54}$$  

What is the largest perfect square that is a factor of 54? 9

Write 54 as a factor of the perfect square and the other factor.

$$\sqrt[]{9\times6}$$  

Simplify

• $$9\sqrt{6}$$  

• $$6\sqrt{3}$$  

• $$6\sqrt{9}$$  

• $$3\sqrt{6}$$  

​1. This method is helpful if you cannot identify perfect square factors.

​2. Factor the number in prime factors. If your number is even, you know you can always start with a factor of 2.

3. Look for pairs of factors​ for square roots, triples for cube roots.​

​4. Pull out every pair from under the radical as a single in front of the radical.

​

5. Drop the pair from inside the radical.​

​

​Simplify Square Roots

Method 2: Prime Factorization​

Multiple Choice

Simplify  Completely $$\sqrt{162}$$  

• $$2\sqrt{9}$$  

• $$9\sqrt{2}$$  

• $$6\sqrt{54}$$  

• $$3\sqrt{54}$$  

Multiple Choice

Simplify $$\sqrt{500}$$  

Use either method.

• $$4\sqrt{125}$$  

• $$5\sqrt{10}$$  

• $$10\sqrt{5}$$  

• $$10\sqrt{50}$$  

Multiple Choice

Simplify $$\sqrt{108}$$  

Use either method.

• $$3\sqrt{6}$$  

• $$6\sqrt{3}$$  

• $$36\sqrt{3}$$  

• $$3\sqrt{36}$$  

Multiple Choice

Simplify $$\sqrt{147}$$  

• impossible

• $$3\sqrt{7}$$  

• cookies

• $$7\sqrt{3}$$