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ACT Math : How to simplify square roots

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Example Questions

Example Question #261 : New Sat Math Calculator

Simplify the following square root:

$\sqrt{83}$

Possible Answers:

$5\sqrt{13}$

$2\sqrt{41.5}$

The square root is already in simplest form.

$4\sqrt{67}$

$8\sqrt{3}$

Correct answer:

The square root is already in simplest form.

Explanation:

We need to factor the number in the square root and find pairs of factors inorder to simplify a square root.

Since 83 is prime, it cannot be factored.

Thus the square root is already simplified.

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Example Question #11 : How To Simplify Square Roots

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

Possible Answers:

$25\sqrt{2}$

$5\sqrt{5}$

$2\sqrt{5}$

$5\sqrt{2}$

Correct answer:

$5\sqrt{2}$

Explanation:

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

a^2 + b^2 = c^2 ---> 5^2 + 5^2 = 50 = c^2

$c= \sqrt50$

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!

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Example Question #53 : Basic Squaring / Square Roots

Simplify: $\sqrt{5000}$

Possible Answers:

$50\sqrt{2}$

$5\sqrt{20}$

$2\sqrt{50}$

$20\sqrt{5}$

$200\sqrt{5}$

Correct answer:

$50\sqrt{2}$

Explanation:

There are two ways to solve this problem. If you happen to have it memorized that 2500 is the perfect square of , then $\sqrt{5000} = \sqrt{2500 \cdot 2} = 50\sqrt 2$ gives a fast solution.

If you haven't memorized perfect squares that high, a fairly fast method can still be achieved by following the rule that any integer that ends in is divisible by , a perfect square.

$\sqrt{5000} = \sqrt{25 \cdot 200} = 5\sqrt{200}$

Now, we can use this rule again:

$5\sqrt{200} = 5\sqrt{25 \cdot 8} = 25 \sqrt {8}$

Remember that we multiply numbers that are factored out of a radical.

The last step is fairly obvious, as there is only one choice:

$25 \sqrt{8} = 25\sqrt {4 \cdot 2} = 50\sqrt 2$

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Example Question #12 : How To Simplify Square Roots

Simplify: $\sqrt{3240}$

Possible Answers:

$18\sqrt{10}$

$15\sqrt{235}$

$9\sqrt{40}$

$35\sqrt{3}$

$9\sqrt{57}$

Correct answer:

$18\sqrt{10}$

Explanation:

A good method for simplifying square roots when you're not sure where to begin is to divide by , or , as one of these generally starts you on the right path. In this case, since our number ends in , let's divide by :

$\sqrt{3240} = \sqrt{324 \cdot 10}$

As it turns out, 324 is a perfect square!

$\sqrt{324 \cdot 10} = 18\sqrt{10}$

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Example Question #13 : How To Simplify Square Roots

Simplify: $8\sqrt{136}$

Possible Answers:

$8\sqrt{136}$

$16\sqrt{8}$

$16\sqrt{34}$

$10\sqrt{34}$

$24\sqrt{3}$

Correct answer:

$16\sqrt{34}$

Explanation:

Again here, if no perfect square is easily recognized try dividing by , , or .

$8\sqrt{136} = 8\sqrt{4 \cdot 34} = 16 \sqrt{34}$

Note that the we obtained by simplifying $\sqrt{4}$ is multiplied, not added, to the already outside the radical.

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Example Question #811 : New Sat

Simplify:

$\sqrt{48}$

Possible Answers:

$4\sqrt3$

$\sqrt{48}$

$3\sqrt{4}$

Correct answer:

$4\sqrt3$

Explanation:

To solve, simply find a perfect square factor and pull it out of the square root.

Recall the factors of 48 include (16, 3). Also recall that 16 is a perfect square since 4*4=16.

Thus,

$\sqrt{48}\Rightarrow \sqrt{16*3}\Rightarrow \sqrt{4^2*3}\Rightarrow 4\sqrt{3}$

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Example Question #11 : Properties Of Roots And Exponents

Solve:

$\sqrt{20}+\sqrt{45}=?$

Possible Answers:

$5\sqrt{5}$

$5\sqrt{20}$

$\sqrt{65}$

$2\sqrt{5}$

$4\sqrt{5}$

Correct answer:

$5\sqrt{5}$

Explanation:

The trick to these problems is to simplify the radical by using the following rule: $\sqrt{ab}=\sqrt{a}*\sqrt{b}$ and $a\sqrt{b}+c\sqrt{b}=(a+c)\sqrt{b}$ Here, we need to find a common factor for the radical. This turns out to be five because $20=5*4\textup{ and }45=9*5.$ Remember, we want to include factors that are perfect squares, which are what nine and four are. Therefore, we can rewrite the equation as: $\sqrt{20}+\sqrt{45}=2\sqrt{5}+3\sqrt{5}=5\sqrt{5}.$

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ACT Math : How to simplify square roots

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #261 : New Sat Math Calculator

Example Question #11 : How To Simplify Square Roots

Example Question #53 : Basic Squaring / Square Roots

Example Question #12 : How To Simplify Square Roots

Example Question #13 : How To Simplify Square Roots

Example Question #811 : New Sat

Example Question #11 : Properties Of Roots And Exponents

All ACT Math Resources

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