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GED Math : Complex Operations

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Example Questions

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Example Question #12 : Square Roots And Radicals

Rationalize: $\frac{6}{5\sqrt6}$

Possible Answers:

$\frac{3\sqrt6}{5}$

$\frac{\sqrt3}{5}$

$\frac{\sqrt6}{30}$

$\frac{2\sqrt6}{5}$

$\frac{\sqrt6}{5}$

Correct answer:

$\frac{\sqrt6}{5}$

Explanation:

Multiply the radical on the top and bottom of the fraction.

$\frac{6}{5\sqrt6} \cdot \frac{\sqrt6}{\sqrt6} = \frac{6\sqrt6}{5(6)}$

Reduce the fraction.

The answer is: $\frac{\sqrt6}{5}$

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Example Question #13 : Square Roots And Radicals

Rationalize: $\frac{5\sqrt2}{\sqrt3}$

Possible Answers:

$\frac{\sqrt{10}}{3}$

$\frac{\sqrt{30}}{3}$

$\frac{5\sqrt6}{3}$

$\frac{3\sqrt6}{5}$

$15\sqrt2$

Correct answer:

$\frac{5\sqrt6}{3}$

Explanation:

Multiply the denominator on the top and bottom.

$\frac{5\sqrt2}{\sqrt3}\cdot \frac{\sqrt3}{\sqrt3} = \frac{5\sqrt6}{3}$

The answer is: $\frac{5\sqrt6}{3}$

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Example Question #13 : Square Roots And Radicals

Simplify the following expression:

$7\sqrt{3}+\sqrt{75}+\sqrt{243}$

Possible Answers:

$21\sqrt{3}$

$\sqrt{3}+5\sqrt{6}$

$14\sqrt{3}+2\sqrt{2}$

$12\sqrt{3}+12$

Correct answer:

$21\sqrt{3}$

Explanation:

Start by analyzing each given term.

$7\sqrt{3}$ cannot be reduced any further, so leave it alone.

Notice that $\sqrt{75}$ can be rewritten as $\sqrt{25(3)}=5\sqrt3$

Next, notice that $\sqrt{243}$ can be rewritten as $\sqrt{(81)(3)}=9\sqrt{3}$

Now, rewrite the original equation:

$7\sqrt{3}+\sqrt{75}+\sqrt{243}=7\sqrt{3}+5\sqrt{3}+9\sqrt{3}=21\sqrt{3}$

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Example Question #21 : Square Roots And Radicals

Simplify the following expression:

$7\sqrt{5}+\sqrt{20}+\sqrt{500}$

Possible Answers:

$19\sqrt5$

$21\sqrt{3}$

$12\sqrt{5}+10\sqrt{2}$

$7\sqrt{5}+4$

Correct answer:

$19\sqrt5$

Explanation:

Start by putting each term in terms of $\sqrt{5}$ .

$7\sqrt{5}$ is already in terms of $\sqrt{5}$ so leave it alone.

Notice that $\sqrt{20}$ can be rewritten as $\sqrt{4\times 5}$ . Thus $\sqrt{20}=2\sqrt{5}$ .

Next, notice that $\sqrt{500}$ can be rewritten as $\sqrt{100\times 5}$ . Thus $\sqrt{500}=10\sqrt{5}$ .

Now, add these terms together.

$7\sqrt{5}+2\sqrt{5}+10\sqrt{5}=19\sqrt{5}$

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Example Question #212 : Complex Operations

Which of the following is true of $\sqrt{792}$ ?

Possible Answers:

$29 < \sqrt{792} < 30$

$30 < \sqrt{792} < 31$

$28 < \sqrt{792} < 29$

$27 < \sqrt{792} < 28$

Correct answer:

$28 < \sqrt{792} < 29$

Explanation:

To determine which two consecutive integers flank $\sqrt{792}$ out of the set $\left \{ 27, 28, 29, 30, 31\right \}$ , square each number. The squares of each number in the set are:

$27^{2} = 27 \times 27 = 729$

$28^{2}= 28 \times 28 = \textbf{784}$

$29^{2} = 29 \times 29 = \textbf{841}$

$30^{2} = 30 \times 30 = 900$

$31^{2} = 31 \times 31 = 961$

784 < 729 < 841 ;

it follows that

$\sqrt{784 }< \sqrt{729 }< \sqrt{841}$

$28 < \sqrt{729 }<29$ .

This is the correct choice.

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Example Question #212 : Complex Operations

Simplify:

$\sqrt{27}+\sqrt{147}+\sqrt{300}$

Possible Answers:

$15\sqrt3$

$20\sqrt5$

$15\sqrt2$

$20\sqrt3$

Correct answer:

$20\sqrt3$

Explanation:

Start by simplifying each radical.

$\sqrt{27}=\sqrt{9\times 3}=3\sqrt{3}$

$\sqrt{147}=\sqrt{49\times 3}=7\sqrt{3}$

$\sqrt{300}=\sqrt{100 \times 3}=10\sqrt{3}$

Now each radical is in terms of $\sqrt{3}$ .

Add them together.

$3\sqrt{3}+7\sqrt{3}+10\sqrt{3}=20\sqrt{3}$

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Example Question #214 : Complex Operations

Simplify:

$\sqrt{28}+\sqrt{63}+\sqrt{112}$

Possible Answers:

$12\sqrt{5}$

$\sqrt{201}$

$9\sqrt{7}$

Correct answer:

$9\sqrt{7}$

Explanation:

Start by simplifying each radical:

$\sqrt{28}=\sqrt{4\times 7}=2\sqrt{7}$

$\sqrt{63}=\sqrt{9\times7}=3\sqrt{7}$

$\sqrt{112}=\sqrt{16 \times 7}=4\sqrt{7}$

Notice that each radical simplifies down into a multiple of $\sqrt{7}$ . Now add up these values to simplify.

$2\sqrt{7}+3\sqrt{7}+4\sqrt{7}=9\sqrt{7}$

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Example Question #215 : Complex Operations

$m = \sqrt{7}$

$n = \sqrt{9}$

$p = \sqrt{11}$

$q= \sqrt{13}$

How many of , , , and are irrational numbers?

Possible Answers:

Correct answer:

Explanation:

The square root of an integer is a rational number if and only if the radicand - the number under the symbol - is itself a perfect square. Of the integers under the four square roots given, only 9 is a perfect square, being equal to $3 \times 3 = 3^{2 }$ . The other three numbers are therefore irrational.

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Example Question #216 : Complex Operations

Simplify $\sqrt{45}$ .

Possible Answers:

$5\sqrt{3}$

$9\sqrt{5}$

$3\sqrt{5}$

Correct answer:

$3\sqrt{5}$

Explanation:

When simplifying a square root, you must break up what's inside the square root into its simplest factors. For example would break up to $2\cdot5$ . Once you do that, you look for pairs of numbers. For each pair, you pull out the common number to the outside of the square root and leave whatever is left over inside the square root. So, for $\sqrt{45}$ , you would break that up to $9\cdot5$ and then $3\cdot3\cdot5$ . As you can see, there is a pair of 3's , so you pull out a and leave whatever is left inside. Since the is left over you would leave that inside the square root. So, you have a outside the square root and a inside the square root, which gives you $3\sqrt{5}$ .

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Example Question #217 : Complex Operations

Solve for $\small z$ : $\small \small \small \sqrt{z}=x^4$

Possible Answers:

$\small \small z=x^2$

$\small z=\sqrt{x^4}$

$\small \small z=\sqrt{x}$

$\small \small z=x^8$

$\small z=2x$

Correct answer:

$\small \small z=x^8$

Explanation:

In order to solve for $\small z$ , we need to move all of the variables on its side of the equation over to the other side of the equation.

We can see that our $\small z$ is hiding in a square root, so in order to get the $\small z$ out we will need to square the whole equation.

$\small \small \small \sqrt{z}=x^4$

$\small (\sqrt{z})^2=(x^4)^2$

Because we're multiply a power of $\small 2$ with the power of $\small 4$ , the two can multiply together to create a power of $\small 8$

$\small z=x^8$

We can't do anything else to this equation as there are no like variables.

Our answer is $\small z=x^8$

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GED Math : Complex Operations

Study concepts, example questions & explanations for GED Math

All GED Math Resources

Example Questions

Example Question #12 : Square Roots And Radicals

Example Question #13 : Square Roots And Radicals

Example Question #13 : Square Roots And Radicals

Example Question #21 : Square Roots And Radicals

Example Question #212 : Complex Operations

Example Question #212 : Complex Operations

Example Question #214 : Complex Operations

Example Question #215 : Complex Operations

Example Question #216 : Complex Operations

Example Question #217 : Complex Operations

All GED Math Resources

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