How to find the square root - SSAT Middle Level Quantitative

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Question

Which of the following numbers has a square root between 18 and 19?

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Answer

For the square root of a number to fall between 18 and 19, the number has to fall between the squares of these numbers - that is:

If , then $18^{2}<x^{2}<19^{2}$ , or, equivalently, $324 <x^{2}<361$

All four choices fall in this range.

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Question

Evaluate: $\sqrt{100} + \sqrt{49 } - \sqrt{36}$

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Answer

Find the individual square roots and perform the operations on them.

$\sqrt{100} = 10$

$\sqrt{49 } = 7$

$\sqrt{36} = 6$

$\sqrt{100} + \sqrt{49 } - \sqrt{36} = 10 + 7 - 6 = 17-6 = 11$

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Question

The square root of a number is 43. What is that number?

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Answer

By definition, the square root of a number multiplied by itself yields that number. Therefore, 43 is the square root of $43 \cdot 43 = 1,849$ .

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Question

The square root of a number is 58. What is that number?

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Answer

By definition, the square root of a number multiplied by itself yields that number. Therefore,

$58 \cdot 58 = 3,364$ .

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Question

$\frac{\sqrt{64}-\sqrt{36}}{\sqrt{4}}=$

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Answer

First, find the square root of each number:

$\sqrt{64}=8$

$\sqrt{36}=6$

$\sqrt{4}=2$

Then, solve the equation accordingly.

$\frac{8-2}{2}=$

$\frac{2}{2}=1$

The answer is

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Question

Evaluate: $\sqrt{121}$

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Answer

The square root of a number is the number which, when multiplied by itself, yields that number. Since $11 \cdot 11 = 121$ , $\sqrt{121} = 11$ .

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Question

Evaluate:

$\sqrt{169}$

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Answer

The square root of a number is the number which, when multiplied by itself, yields that number. Since $13 \cdot 13 = 169$ , $\sqrt{169} = 13$ .

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Question

Which of the following statements is true about $\sqrt{441}$ ?

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Answer

The square root of a number is the number which, when multiplied by itself, yields that number. Since $21 \cdot 21 = 441$ , $\sqrt{441} = 21$ .

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Question

Which of the following statements is true about $\sqrt{ 231}$ ?

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Answer

The square root of a number is the number which, when multiplied by itself, yields that number. Since $15 \cdot 15 = 225$ , $16 \cdot 16 = 256$ , and

$15^{2} = 225 < 231 < 256 = 16^{2}$

then

$15 = \sqrt{225 }< 231 <\sqrt{ 256} = 16$

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Question

A square has an area of $16\ ft^{2}$ .

How long is each of its sides?

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Answer

The two sides of the square must be the same length, and multiply to give 16. So we are looking for the $\sqrt{16}$ which is .

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Question

Which of the following statements is true about $\sqrt{-225}$ ?

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Answer

A negative number does not have a real square root, so this is the correct choice.

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Question

Evaluate:

$\sqrt{121} + \sqrt{49}$

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Answer

$11 ^{2 } = 11 \times 11 = 121$ , so $\sqrt{121} = 11$ .

$7 ^{2 } = 7 \times 7 = 49$ , so $\sqrt{49} = 7$ .

$\sqrt{121} + \sqrt{49} = 11 + 7 = 18$

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Question

Let $N = \sqrt{10 \times 20 + 7 \times 8}$

Which of the following is a true statement?

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Answer

$N = \sqrt{10 \times 20 + 7 \times 8}= \sqrt{200 + 56}= \sqrt{256}$

Since , $N = \sqrt{256} = 16$

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Question

Let $N = \sqrt{7^{2}-5^{2}-3^{2}}$

Then which of the following statements is correct?

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Answer

$N = \sqrt{7^{2}-5^{2}-3^{2}} = \sqrt{49-25-9} = \sqrt{24-9} = \sqrt{15}$

, so

$\sqrt{9} < \sqrt{15} < \sqrt{16}$

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Question

$x^{2} = 49, \left | x\right | + 7 = ?$

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Answer

If $x^{2} = 49$ , then $x=\pm 7$ .

$\left | 7 \right |=7,\ \left | -7 \right |=7$

Therefore, $\left | x \right |+7=7+7=14$ .

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Question

Which of the answer choices is equivalent to $2\sqrt{3}$ ?

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Answer

can also be written as $\sqrt{4}$ , so $2\sqrt{3}$ can also be written as $\sqrt{4\cdot 3}$ , or $\sqrt{12}$ .

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Question

Which of the answer choices is equivalent to $\sqrt{24}$ ?

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Answer

Recognize that $\sqrt{24}=\sqrt{4\cdot 6}$ . Since $\sqrt{4}=2$ , we can move the outside of the radical sign. This leaves us with $2\sqrt{6}$ .

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Question

What is the square root of ?

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Answer

The square root of is , since $7\cdot7=49$ .

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Question

Evaluate:

$\sqrt{\frac{169}{100}}$

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Answer

To find the square root of a fraction, extract the square root of both the numerator and the denominator. Since $10 ^{2} = 10 \times 10 = 100$ , $\sqrt{100} = 10$ , and since $13^{2} = 169$ , $\sqrt{169} = 13$ .

Combine these results:

$\sqrt{\frac{169}{100}} = \frac{\sqrt{169}}{\sqrt{100}} = \frac{13}{10}$

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Question

Evaluate:

$\sqrt{0.0009}$

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Answer

$0.0009= \frac{9}{10,000}$ , so

$\sqrt{0.0009} = \sqrt{\frac{9}{10,000}}$ .

The square root of a fraction can be determined by taking the square roots of both numerator and denominator. Since $3^{2} = 9$ , $\sqrt{9 } = 3$ , and since $100^{2} = 10,000$ , $\sqrt{10,000} = 100$ . Therefore,

$\sqrt{0.0009} = \sqrt{\frac{9}{10,000}} = \frac{\sqrt{9}}{\sqrt{10,000}} = \frac{3}{100} = 0.03$ .

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