ACT Math - Basic Squaring / Square Roots

Simplify the following expression: $\sqrt{60}+\sqrt{40}+\sqrt{10}$

$2\sqrt{15}+3\sqrt{10}$

$5\sqrt{25}$

$2\sqrt{15}+2\sqrt{20}$

$4\sqrt{15}+5\sqrt{5}$

Explanation

Begin by factoring out each of the radicals:

$\sqrt{60}+\sqrt{40}+\sqrt{10}=\sqrt{4*15}+\sqrt{4*10}+\sqrt{10}$

For the first two radicals, you can factor out a $\sqrt{4}$ or :

$2\sqrt{15}+2\sqrt{10}+\sqrt{10}$

The other root values cannot be simply broken down. Now, combine the factors with $\sqrt{10}$ :

$2\sqrt{15}+2\sqrt{10}+\sqrt{10}=2\sqrt{15}+3\sqrt{10}$

This is your simplest form.

Simplify the following expression: $\sqrt{60}+\sqrt{40}+\sqrt{10}$

$2\sqrt{15}+3\sqrt{10}$

$5\sqrt{25}$

$2\sqrt{15}+2\sqrt{20}$

$4\sqrt{15}+5\sqrt{5}$

Explanation

Begin by factoring out each of the radicals:

$\sqrt{60}+\sqrt{40}+\sqrt{10}=\sqrt{4*15}+\sqrt{4*10}+\sqrt{10}$

For the first two radicals, you can factor out a $\sqrt{4}$ or :

$2\sqrt{15}+2\sqrt{10}+\sqrt{10}$

The other root values cannot be simply broken down. Now, combine the factors with $\sqrt{10}$ :

$2\sqrt{15}+2\sqrt{10}+\sqrt{10}=2\sqrt{15}+3\sqrt{10}$

This is your simplest form.

Simplify the following expression: $\sqrt{60}+\sqrt{40}+\sqrt{10}$

$2\sqrt{15}+3\sqrt{10}$

$5\sqrt{25}$

$2\sqrt{15}+2\sqrt{20}$

$4\sqrt{15}+5\sqrt{5}$

Explanation

Begin by factoring out each of the radicals:

$\sqrt{60}+\sqrt{40}+\sqrt{10}=\sqrt{4*15}+\sqrt{4*10}+\sqrt{10}$

For the first two radicals, you can factor out a $\sqrt{4}$ or :

$2\sqrt{15}+2\sqrt{10}+\sqrt{10}$

The other root values cannot be simply broken down. Now, combine the factors with $\sqrt{10}$ :

$2\sqrt{15}+2\sqrt{10}+\sqrt{10}=2\sqrt{15}+3\sqrt{10}$

This is your simplest form.

Simplify: $\small \small \sqrt{140} + \sqrt{315}$ .

$\small 5\sqrt{35}$

$\small \sqrt{455}$

$\small 21\sqrt{3}$

$\small 5\sqrt{70}$

$\textup{This is the simplest possible form}$

Explanation

Begin simplifying by breaking apart the square roots in question. Thus, you know:

$\small \sqrt{140} + \sqrt{315} = \sqrt{4*35} + \sqrt{9 * 35} = 2\sqrt{35} + 3\sqrt{35}$

Now, with square roots, you can combine factors just as if a given root were a variable. So, just as $\small 2x + 3x = 5x$ , so too does $\small 2\sqrt{35} + 3\sqrt{35} = 5\sqrt{35}$ .

Simplify: $\small \small \sqrt{140} + \sqrt{315}$ .

$\small 5\sqrt{35}$

$\small \sqrt{455}$

$\small 21\sqrt{3}$

$\small 5\sqrt{70}$

$\textup{This is the simplest possible form}$

Explanation

Begin simplifying by breaking apart the square roots in question. Thus, you know:

$\small \sqrt{140} + \sqrt{315} = \sqrt{4*35} + \sqrt{9 * 35} = 2\sqrt{35} + 3\sqrt{35}$

Now, with square roots, you can combine factors just as if a given root were a variable. So, just as $\small 2x + 3x = 5x$ , so too does $\small 2\sqrt{35} + 3\sqrt{35} = 5\sqrt{35}$ .

Simplify: $\small \small \sqrt{140} + \sqrt{315}$ .

$\small 5\sqrt{35}$

$\small \sqrt{455}$

$\small 21\sqrt{3}$

$\small 5\sqrt{70}$

$\textup{This is the simplest possible form}$

Explanation

Begin simplifying by breaking apart the square roots in question. Thus, you know:

$\small \sqrt{140} + \sqrt{315} = \sqrt{4*35} + \sqrt{9 * 35} = 2\sqrt{35} + 3\sqrt{35}$

Now, with square roots, you can combine factors just as if a given root were a variable. So, just as $\small 2x + 3x = 5x$ , so too does $\small 2\sqrt{35} + 3\sqrt{35} = 5\sqrt{35}$ .

Right triangle $\Delta ABC$ has legs of length . What is the exact length of the hypotenuse?

$5\sqrt{2}$

$25\sqrt{2}$

$5\sqrt{5}$

$2\sqrt{5}$

Explanation

If the triangle is a right triangle, then it follows the Pythagorean Theorem. Therefore:

--->

$c= \sqrt50$

At this point, factor out the greatest perfect square from our radical:

$\sqrt{50} = \sqrt{25 \cdot 2}$

Simplify the perfect square, then repeat the process if necessary.

$\sqrt{25 \cdot 2} = 5\sqrt2$

Since is a prime number, we are finished!

Simplify the following:

$\small \sqrt{132} - \sqrt{297}$

$-\sqrt{33}$

$\small -\sqrt{165}$

$-10\sqrt{21}$

$-2\sqrt{99}$

$\textup{The answer is not a real number}$

Explanation

Begin simplifying by breaking apart the square roots in question. Thus, you know:

$\small \small \small \sqrt{132} - \sqrt{297} = \sqrt{4*33} - \sqrt{9 * 33} = 2\sqrt{33} - 3\sqrt{33}$

Now, with square roots, you can combine factors just as if a given root were a variable. So, just as $\small 2x-3x=-x$ , so too does $\small 2\sqrt{33} - 3\sqrt{33}=-\sqrt{33}$ .

Simplify the following:

$\small \sqrt{132} - \sqrt{297}$

$-\sqrt{33}$

$\small -\sqrt{165}$

$-10\sqrt{21}$

$-2\sqrt{99}$

$\textup{The answer is not a real number}$

Explanation

Begin simplifying by breaking apart the square roots in question. Thus, you know:

$\small \small \small \sqrt{132} - \sqrt{297} = \sqrt{4*33} - \sqrt{9 * 33} = 2\sqrt{33} - 3\sqrt{33}$

Now, with square roots, you can combine factors just as if a given root were a variable. So, just as $\small 2x-3x=-x$ , so too does $\small 2\sqrt{33} - 3\sqrt{33}=-\sqrt{33}$ .

Simplify the following:

$\small \sqrt{132} - \sqrt{297}$

$-\sqrt{33}$

$\small -\sqrt{165}$

$-10\sqrt{21}$

$-2\sqrt{99}$

$\textup{The answer is not a real number}$

Explanation

Begin simplifying by breaking apart the square roots in question. Thus, you know:

$\small \small \small \sqrt{132} - \sqrt{297} = \sqrt{4*33} - \sqrt{9 * 33} = 2\sqrt{33} - 3\sqrt{33}$

Now, with square roots, you can combine factors just as if a given root were a variable. So, just as $\small 2x-3x=-x$ , so too does $\small 2\sqrt{33} - 3\sqrt{33}=-\sqrt{33}$ .

Basic Squaring / Square Roots

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