Factoring and Simplifying Square Roots - PSAT Math

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

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

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

x$\sqrt{9}$$\sqrt{5}$+x$\sqrt{9}$$\sqrt{4times 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{3sqrt{2}$}{3$\sqrt{5}$+6$\sqrt{2}$}

Simplifying, this becomes

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

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

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

√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

4√27 + 16√75 +3√12 =

4*(√3)*(√9) + 16*(√3)*(√25) +3*(√3)*(√4) =

4*(√3)*(3) + 16*(√3)*(5) + 3*(√3)*(2) =

12√3 + 80√3 +6√3= 98√3

First we will list the factors of 3888:

$3888=3times1296=3times3times432=3^2times12times36$$=3^2times12times12times3$$=3^2times12$^2times3

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 .

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

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.

To simplify a square root, you can break the number down into its prime factors using a factor tree. The prime factors of 72 are . Let's take each piece separately.

The square root of can be simplified to be which is the same as .

The square root of is .

When you multiply together your answers,

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

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

√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

4√27 + 16√75 +3√12 =

4*(√3)*(√9) + 16*(√3)*(√25) +3*(√3)*(√4) =

4*(√3)*(3) + 16*(√3)*(5) + 3*(√3)*(2) =

12√3 + 80√3 +6√3= 98√3

First we will list the factors of 3888:

$3888=3times1296=3times3times432=3^2times12times36$$=3^2times12times12times3$$=3^2times12$^2times3

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 .

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

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.