PSAT Math : Factoring and Simplifying Square Roots

Study concepts, example questions & explanations for PSAT Math

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Example Questions

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Example Question #1 : Factoring And Simplifying Square Roots

Solve for \dpi{100} x:

x\sqrt{45}+x\sqrt{72}=\sqrt{18}

Possible Answers:

x=3

x=\frac{\sqrt{5}}{\sqrt{2}}+2

x=\sqrt{9}

x=\frac{\sqrt{2}}{\sqrt{5}+2\sqrt{2}}

x=\frac{\sqrt{2}}{\sqrt{5}}+\frac{1}{2}

Correct answer:

x=\frac{\sqrt{2}}{\sqrt{5}+2\sqrt{2}}

Explanation:

x\sqrt{45}+x\sqrt{72}=\sqrt{18}

Notice how all of the quantities in square roots are divisible by 9

x\sqrt{9\times 5}+x\sqrt{9\times 8}=\sqrt{9\times 2}

x\sqrt{9}\sqrt{5}+x\sqrt{9}\sqrt{4\times 2}=\sqrt{9}\sqrt{2}

3x\sqrt{5}+3x\sqrt{4}\sqrt{2}=3\sqrt{2}

3x\sqrt{5}+6x\sqrt{2}=3\sqrt{2}

x(3\sqrt{5}+6\sqrt{2})=3\sqrt{2}

x=\frac{3\sqrt{2}}{3\sqrt{5}+6\sqrt{2}}

Simplifying, this becomes

x=\frac{\sqrt{2}}{\sqrt{5}+2\sqrt{2}}

Example Question #51 : Simplifying Square Roots

If m and n are postive integers and 4m = 2n, what is the value of m/n?

Possible Answers:

4

1/2

8

2

16

Correct answer:

1/2

Explanation:
  1. 2= 4. Also, following the rules of exponents, 4= 1. 
  2. One can therefore say that m = 1 and n = 2.
  3. The question asks to solve for m/n. Since m = 1 and n = 2, m/n = 1/2.

Example Question #1 : Factoring And Simplifying Square Roots

Simplify the radical:

Possible Answers:

Correct answer:

Explanation:

 

Example Question #2 : Factoring And Simplifying Square Roots

Simplify. Assume all variables are positive real numbers. 

Possible Answers:

Correct answer:

Explanation:

The index coefficent in is represented by . When no index is present, assume it is equal to 2.  under the radical is known as the radican, the number you are taking a root of. 

First look for a perfect square, 

Then to your Variables 

Take your exponents on both variables and determine the number of times our index will evenly go into both. 

So you would take out a  and would be left with a 

*Dividing the radican exponent by the index - gives you the number of variables that should be pulled out.

The final answer would be .

Example Question #3 : Factoring And Simplifying Square Roots

Simplify. Assume all integers are positive real numbers. 

Possible Answers:

Correct answer:

Explanation:

Index of means the cube root of Radican 

Find a perfect cube in    

Simplify the perfect cube, giving you .

Take your exponents on both variables and determine the number of times our index will evenly go into both.

 


The final answer would be

Example Question #1 : How To Simplify Square Roots

Simplify square roots. Assume all integers are positive real numbers. 

Simplify as much as possible. List all possible answers.

1a.

1b. 

1c. 

Possible Answers:

 and  and

 and 

 and  and 

 and  and 

Correct answer:

 and  and 

Explanation:

When simplifying radicans (integers under the radical symbol), we first want to look for a perfect square. For example, is not a perfect square. You look to find factors of  to see if there is a perfect square factor in , which there is.

1a. 

Do the same thing for .

1b.

1c.Follow the same procedure except now you are looking for perfect cubes. 

Example Question #1 : How To Simplify Square Roots

Simplify

÷ √3

Possible Answers:

not possible

3

none of these

2

3√3

Correct answer:

3√3

Explanation:

in order to simplify a square root on the bottom, multiply top and bottom by the root

Asatsimplifysquare_root

Example Question #1 : How To Simplify Square Roots

Simplify:

√112

Possible Answers:

10√12

4√10

20

4√7

12

Correct answer:

4√7

Explanation:

√112 = {√2 * √56} = {√2 * √2 * √28} = {2√28} = {2√4 * √7} = 4√7 

Example Question #3 : Factoring And Simplifying Square Roots

Simplify:

 

√192

Possible Answers:
4√2
8√3
None of these
8√2
4√3
Correct answer: 8√3
Explanation:

√192 = √2 X √96 

√96 = √2 X √48

√48 = √4 X√12

√12 = √4 X √3

√192 = √(2X2X4X4) X √3

        = √4X√4X√4  X √3

        = 8√3

Example Question #1 : How To Simplify Square Roots

What is the simplest way to express \sqrt{3888}?

Possible Answers:

2\sqrt{972}

2304\sqrt{2}

12\sqrt{27}

144\sqrt{27}

Correct answer:

Explanation:

First we will list the factors of 3888:

3888=3\times1296=3\times\3\times432=3^2\times12\times36=3^2\times12\times12\times3=3^2\times12^2\times3

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