Square Roots and Operations - SAT Math

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Question

Simplify in radical form:

$\sqrt{507}+\sqrt{12}$

Answer

To simplify, break down each square root into its component factors:

$\sqrt{507}+\sqrt{12}$

$\sqrt{169\cdot 3}+\sqrt{4\cdot 3}$

$\sqrt{13\cdot13\cdot3}:+\sqrt{2\cdot 2\cdot 3}$

You can remove pairs of factors and bring them outside the square root sign. At this point, since each term shares $\sqrt3$ , you can add them together to yield the final answer:

$13\sqrt{3}+2\sqrt{3}=15\sqrt{3}$

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Question

Simplify: $\frac{2\sqrt{3}}{\sqrt{2}}+\frac{4\sqrt{2}}{\sqrt{3}}$

Answer

Take each fraction separately first:

(2√3)/(√2) = \[(2√3)/(√2)\] * \[(√2)/(√2)\] = (2 * √3 * √2)/(√2 * √2) = (2 * √6)/2 = √6

Similarly:

(4√2)/(√3) = \[(4√2)/(√3)\] * \[(√3)/(√3)\] = (4√6)/3 = (4/3)√6

Now, add them together:

√6 + (4/3)√6 = (3/3)√6 + (4/3)√6 = (7/3)√6

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Question

If $\sqrt{x}=3^2$ what is $x$ ?

Answer

Square both sides:

x = (32)2 = 92 = 81

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Question

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

Answer

Begin by factoring out each of the radicals:

$\sqrt{60}+\sqrt{40}+\sqrt{10}=\sqrt{415}+\sqrt{410}+\sqrt{10}$

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.

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Question

Solve for .

Note, $x\ge0$ :

$15\sqrt{x}-10=4\sqrt{x}+4$

Answer

Begin by getting your terms onto the left side of the equation and your numeric values onto the right side of the equation:

$15\sqrt{x}-4\sqrt{x}=4+10$

Next, you can combine your radicals. You do this merely by subtracting their respective coefficients:

$11\sqrt{x}=14$

Now, square both sides:

$(11\sqrt{x})^2=(14)^2$

$(11)^2(\sqrt{x})^2=(14)^2$

Solve by dividing both sides by :

$x=\frac{196}{121}$

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Question

(√27 + √12) / √3 is equal to

Answer

√27 is the same as 3√3, while √12 is the same as 2√3.

3√3 + 2√3 = 5√3

(5√3)/(√3) = 5

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Question

Simplify: $\frac{\sqrt{210}}{\sqrt{6}}$

Answer

When dividing square roots, we divide the numbers inside the radical. Simplify if necessary.

$\frac{\sqrt{210}}{\sqrt{6}}=\frac{\sqrt{35}*\sqrt{6}}{\sqrt{6}}=\sqrt{35}$

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Question

Simplify: $\frac{\sqrt{432}}{\sqrt{36}}$

Answer

When dividing square roots, we divide the numbers inside the radical. Simplify if necessary.

$\frac{\sqrt{432}}{\sqrt{36}}=\frac{\sqrt{36}\sqrt{12}}{\sqrt{36}}=\sqrt{12}$

Let's simplify this even further by factoring out a .

$\sqrt{12}=\sqrt{4}\sqrt{3}=2\sqrt{3}$

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Question

Simplify: $\frac{\sqrt{1200}}{\sqrt{8}}$

Answer

When dividing square roots, we divide the numbers inside the radical. Simplify if necessary.

$\frac{\sqrt{1200}}{\sqrt{8}}=\frac{\sqrt{8}\sqrt{150}}{\sqrt{8}}=\sqrt{150}$
Let's simplify this even further by factoring out a .

$\sqrt{150}=\sqrt{25}\sqrt{6}=5\sqrt{6}$

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Question

Simplify: $\sqrt{7}*\sqrt{14}$

Answer

When multiplying square roots, you are allowed to multiply the numbers inside the square root. Then simplify if necessary.

$\sqrt{7}\sqrt{14}=\sqrt{714}=\sqrt{98}=\sqrt{49}*\sqrt{2}=7\sqrt{2}$

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Question

Simplify: $\sqrt{10}*\sqrt{15}$

Answer

When multiplying square roots, you are allowed to multiply the numbers inside the square root. Then simplify if necessary.

$\sqrt{10}\sqrt{15}=\sqrt{1015}=\sqrt{150}=\sqrt{25}*\sqrt{6}=5\sqrt{6}$

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Question

Simplify: $\sqrt{24}*3\sqrt{8}$

Answer

When multiplying square roots, you are allowed to multiply the numbers inside the square root. Then simplify if necessary.

$\sqrt{24}3\sqrt{8}=3\sqrt{248}=3\sqrt{192}=3\sqrt{64}*\sqrt{3}=24\sqrt{3}$

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Question

Evaluate and simplify:

$\sqrt{10}*\sqrt{12}$

Answer

To multiply square roots, we multiply the numbers inside the radical and we can simplify them if possible.

$\ \sqrt{10}\sqrt{12}\=\sqrt{1012}\=\sqrt{120}\=\sqrt{4}*\sqrt{30}\=2\sqrt{30}$

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Question

Simplify and evaluate:

$\sqrt{12}*\sqrt{48}$

Answer

To multiply square roots, we multiply the numbers inside the radical and we can simplify them if possible.

In this case, let's simplify each individual radical and multiply them.

$\\sqrt{12}\sqrt{48}\=\sqrt{4}\sqrt{3}\sqrt{16}\sqrt{3}\=2\sqrt{3}4\sqrt{3}\=83=24$

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Question

Simplify: $\sqrt{x+5}*\sqrt{x-5}$

Answer

To multiply square roots, we multiply the numbers inside the radical and we can simplify them if possible.

$\\sqrt{x+5}*\sqrt{x-5}\=\sqrt{(x+5)(x-5)}\=\sqrt{x^2-25}$

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Question

Simplify:

$\sqrt{6}*2\sqrt{3}$

Answer

To multiply square roots, we multiply the numbers inside the radical.

Any numbers outside the radical are also multiplied. We can simplify them if possible.

$\\sqrt{6}2\sqrt{3}\=2\sqrt{63}\=2\sqrt{18}\=2\sqrt{9}\sqrt{2}\=23\sqrt{2}\=6\sqrt{2}$

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Question

Simplify:

$4\sqrt{7}*\sqrt{21}$

Answer

To multiply square roots, we multiply the numbers inside the radical.

Any numbers outside the radical are also multiplied.

We can simplify them if possible.

$\4\sqrt{7}\sqrt{21}\ =4\sqrt{721}\=4\sqrt{773}\=4*7\sqrt{3}\=28\sqrt{3}$

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Question

Simplify:

$2\sqrt{10}*5\sqrt{20}$

Answer

To multiply square roots, we multiply the numbers inside the radical.

Any numbers outside the radical are also multiplied.

We can simplify them if possible.

$\2\sqrt{10}5\sqrt{20}\=10\sqrt{1020}\=10\sqrt{10102}\=10*10\sqrt{2}\=100\sqrt{2}$

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Question

Simplify:

$\sqrt{3}(\sqrt{3}+4)$

Answer

To simplify the problem, just distribute the radical to each term in the parentheses.

$\\sqrt{3}(\sqrt{3}+4)\=\sqrt{9}+4\sqrt{3}\=3+4\sqrt{3}$

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Question

Simplify:

$\sqrt{3}(\sqrt{7}+\sqrt{12})$

Answer

To simplify the problem, just distribute the radical to each term in the parentheses.

$\\sqrt{3}(\sqrt{7}+\sqrt{12})\=\sqrt{21}+\sqrt{36}\=\sqrt{21}+6$

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