Factoring and Simplifying Square Roots - PSAT Math

Card 0 of 12

Question

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

Answer

22 = 4. Also, following the rules of exponents, 41 = 1.

One can therefore say that m = 1 and n = 2.

The question asks to solve for m/n. Since m = 1 and n = 2, m/n = 1/2.

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Question

Solve for $\dpi{100} x$ :

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

Answer

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

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

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

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Question

Simplify the radical:

$\sqrt{320}$

Answer

$\sqrt{320}=\sqrt{2\times 160}=\sqrt{2\times 2\times 80}=2\sqrt{80}=2\sqrt{2\times 40}=2\sqrt{2\times 2\times 20}=2\times 2\times \sqrt{20}=4\sqrt{2\times 10}=4\sqrt{2\times 2\times 5}=4\times 2\times \sqrt{5}=8\sqrt{5}$

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Question

Simplify

9 ÷ √3

Answer

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

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Question

Simplify:

√112

Answer

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

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Question

Simplify:

√192

Answer

√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

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Question

Simplify:

$4\sqrt{27}+16\sqrt{75}+3\sqrt{12}$

Answer

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

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Question

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

Answer

First we will list the factors of 3888:

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

$\sqrt{3888}=\sqrt{3^{2}\times 12^{2}\times 3}=\sqrt{3^{2}}\times\sqrt{12^{2}} \times \sqrt{3}=3\times 12\times \sqrt{3}=36\sqrt{3}$

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Question

Simplify. Assume all variables are positive real numbers.

$\sqrt{16x^{5}{y^{2}}}$

Answer

$\sqrt{16x^{5}{y^{2}}}$

The index coefficent in $\sqrt[n]{b}$ 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, $\sqrt{16}=4$

Then to your Variables $\sqrt{x^{5}y^{2}}$

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

$\sqrt{x^{5}} \rightarrow 2\cdot2=4 +1$

So you would take out a $x^{2}$ and would be left with a $\sqrt{x}$

$\sqrt{y^{2}} \rightarrow 2\cdot1=2$

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

The final answer would be $4x^{2}y\sqrt{x}$ .

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Question

Simplify. Assume all integers are positive real numbers.

$\sqrt[3]{24x^{16}y^{12}}$

Answer

$\sqrt[3]{24x^{16}y^{12}}$

Index of means the cube root of Radican $24x^{16}y^{12}$

Find a perfect cube in $\rightarrow$ $\sqrt[3]{8\cdot 3}$

Simplify the perfect cube, giving you $2\sqrt[3]{3}$ .

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

$x\rightarrow \frac{16}{3}= 5^{\frac{1}{3}} \rightarrow x^{5}\sqrt[3]{x}$

$y\rightarrow \frac{12}{3}= 4 \rightarrow y^{4}$

The final answer would be

$2x^{5}y^{4}\sqrt[3]{3x}$

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Question

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

Simplify as much as possible. List all possible answers.

1a. $\sqrt{12}$

1b. $\sqrt{48}$

1c. $\sqrt[3]{24}$

Answer

When simplifying radicans (integers under the radical symbol), we first want to look for a perfect square. For example, $\sqrt{12}$ 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. $\sqrt{12}=\sqrt{4\cdot 3}=\sqrt{4}\cdot\sqrt{3}= 2\sqrt{3}$

Do the same thing for $\sqrt{48}$ .

1b. $\sqrt{48}=\sqrt{16\cdot 3}=\sqrt{16}\cdot\sqrt{3}=4\sqrt{3}$

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

$\sqrt[3]{24}=\sqrt[3]{8\cdot 3}=\sqrt[3]{8}\cdot\sqrt[3]{3}=2\sqrt[3]{3}$

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Question

Simplify: $\sqrt{72}$

Answer

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 $2^{3}\times3^{2}$ . Let's take each piece separately.

The square root of $2^{3}$ can be simplified to be $\sqrt{2^{2}}\times \sqrt{2^{1}}$ which is the same as $2\sqrt{2}$ .

The square root of $3^{2}$ is .

When you multiply together your answers, $2\sqrt{2}\times3=6\sqrt{2}$

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