How to add square roots - SAT Math

Card 0 of 5

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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