Simplifying Square Roots

Simplifying Square Roots

Unit 1

Simplifying vs Estimating

When simplifying an equation/value, you are changing it into the most simple/easy form. This is an exact value.
When estimating an equation/value, you are finding a value that the equation is almost equal to. This is not an exact value.
Example with a fraction : 
 $\frac{60}{22}\ \ \ \ Simplify\ :\ 2\ \frac{8}{11}\ or\ \frac{30}{11}\ \ Estimate\ :\ 3$  

Example with a square root: 

 $\sqrt{32}\ \ \ \ Simplify\ :\ 4\sqrt{2}\ \ \ \ \ \ \ \ \ Estimate\ :\ 5.5$  

Simplifying Square Roots

1. Find a factor of the number in the root that is a square of a number.
Ex : 72 has a factor of 36
8 has a factor of 4

2. Take the square root of that factor out and place it in front of the square root.

 $\sqrt{72}=\sqrt{36\times2}=\sqrt{6^2\times2}=6\sqrt{2}$  

Factoring Tips

1. If the sum of each digit in a number is equal to a multiple of 3, 3 is a factor of that number.

Examples :

453 ; 4+5+3=9 ; 9 is a multiple of 3 so 3 is a factor of 453. (3, 151)

5730 ; 5+7+3+0=15 ; 15 is a multiple of 3 so 3 is a factor of 5730 (3, 1910)

2. If the number is even, it has a factor of 2.

3. If a number has a factor of 2 and 3, it will also have a factor of 6.

Factoring Practice

Open Ended

What are the factors of 84?

Solution

84

1 ; 84

2 ; 42

3 ; 28

4 ; 21

6 ; 14

7 ; 12

1, 2, 3, 4, 5, 6, 7, 12, 14 ,21 ,28, 42, 84

Open Ended

What are the factors of 162?

Solution

162

1 ; 162

2 ; 81

3 ; 54

6 ; 27

9 ; 18

1, 2, 3, 6, 9, 18, 27, 54, 81, 162

Open Ended

What are the factors of 92?

What factor(s) of 92 is/are squares of a number?

Solution

Factors of 92

1 ; 92

2 ; 46

4 ; 23

1, 2, 4, 23, 46, 92

4 is a square of 2.

Simplifying Square Roots

 $\sqrt{92}=\sqrt{4\times23}=\sqrt{2^2\times23}=2\sqrt{23}$  

1 : Find a factor that is a square. The quickest to check are 4, 9, 25, 36,

2: Separate the number into the square factor times it's pair. 

3: Take the square root of the square factor out.

4. If it can still be simplified repeat the steps above.

Simplifying Square Roots Practice

Open Ended

 $Simplify\ \sqrt{75}$  

Solution

 $\sqrt{75}=\sqrt{25\times3}=\sqrt{5^2\times3}=5\sqrt{3}$  

Open Ended

 $Simplify\ \sqrt{54}$  

Solution

 $\sqrt{54}=\sqrt{9\times6}=\sqrt{3^2\times6}=3\sqrt{6}$