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ISEE Middle Level Quantitative : How to find the square root

Study concepts, example questions & explanations for ISEE Middle Level Quantitative

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All ISEE Middle Level Quantitative Resources

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Example Questions

Example Question #1 : How To Find The Square Root

Give the square root of 256.

Possible Answers:

$\displaystyle 14$

$\displaystyle 18$

$\displaystyle 24$

$\displaystyle 22$

$\displaystyle 16$

Correct answer:

$\displaystyle 16$

Explanation:

$16\cdot16= 256$ $\displaystyle 16\cdot16= 256$ - that is, 16 squared is 256, making 16 the square root of 256 by definition.

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Example Question #2 : How To Find The Square Root

Which is the greater quantity?

(a) $\sqrt{425 }$ $\displaystyle \sqrt{425 }$

(b) $\displaystyle 21$

Possible Answers:

(a) and (b) are equal

(b) is greater

It is impossible to tell from the information given

(a) is greater

Correct answer:

(b) is greater

Explanation:

$21^{2} = 21 \times 21 = 441$ $\displaystyle 21^{2} = 21 \times 21 = 441$ , so $21 = \sqrt{441}$ $\displaystyle 21 = \sqrt{441}$

$\displaystyle 441 > 425$ , so

$21 = \sqrt{441} > \sqrt{425}$ $\displaystyle 21 = \sqrt{441} > \sqrt{425}$

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Example Question #1 : How To Find The Square Root

Which is the greater quantity?

(a) $6 \sqrt{ 7}$ $\displaystyle 6 \sqrt{ 7}$

(b) $7 \sqrt{6}$ $\displaystyle 7 \sqrt{6}$

Possible Answers:

(a) is greater

(a) and (b) are equal

(b) is greater

It is impossible to tell from the information given

Correct answer:

(b) is greater

Explanation:

(a) $\left (6 \sqrt{7} \right )^{2} = 6^{2}\left ( \sqrt{7} \right ) ^{2} = 36 \cdot 7 = 252$ $\displaystyle \left (6 \sqrt{7} \right )^{2} = 6^{2}\left ( \sqrt{7} \right ) ^{2} = 36 \cdot 7 = 252$ , so $6 \sqrt{7} = \sqrt{ 252 }$ $\displaystyle 6 \sqrt{7} = \sqrt{ 252 }$

(b) $\left (7 \sqrt{6} \right )^{2} = 7^{2}\left ( \sqrt{6} \right ) ^{2} = 49 \cdot 6 = 294$ $\displaystyle \left (7 \sqrt{6} \right )^{2} = 7^{2}\left ( \sqrt{6} \right ) ^{2} = 49 \cdot 6 = 294$ , so $7\sqrt{6} = \sqrt{ 294 }$ $\displaystyle 7\sqrt{6} = \sqrt{ 294 }$

$\displaystyle 294 > 252$ ,

$\sqrt{294 }> \sqrt{252}$ $\displaystyle \sqrt{294 }> \sqrt{252}$ ,

and

$7\sqrt{6} > 6 \sqrt{7}$ $\displaystyle 7\sqrt{6} > 6 \sqrt{7}$

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Example Question #2 : How To Find The Square Root

Column A Column B

$\sqrt{100}$ $\displaystyle \sqrt{100}$ $\sqrt{10.0}$ $\displaystyle \sqrt{10.0}$

Possible Answers:

The quantity in Column A is greater.

The quantity in Column B is greater.

There is no relationship between the quantities.

The quantities are equal.

Correct answer:

The quantity in Column A is greater.

Explanation:

The square root of 100 is 10, while the square root of 10 is between the square root of 9 and 16, so about 4. Therefore, Column A has to be greater.

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Example Question #1 : Squares / Square Roots

Which of the following is equal to $\sqrt{5^{2}+7^{2}+7}$ $\displaystyle \sqrt{5^{2}+7^{2}+7}$ ?

Possible Answers:

$\displaystyle 9$

$\displaystyle 11$

$\displaystyle 10$

$\displaystyle 8$

None of the other responses is correct.

Correct answer:

$\displaystyle 9$

Explanation:

$\sqrt{5^{2}+7^{2}+7}$ $\displaystyle \sqrt{5^{2}+7^{2}+7}$

$=\sqrt{25 +49+7}$ $\displaystyle =\sqrt{25 +49+7}$

$=\sqrt{81}$ $\displaystyle =\sqrt{81}$

$\displaystyle =9$

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Example Question #6 : How To Find The Square Root

Which of the following is equal to $\sqrt{11^{2}+ 6 \cdot 8}$ $\displaystyle \sqrt{11^{2}+ 6 \cdot 8}$ ?

Possible Answers:

$\displaystyle 14$

$\displaystyle 13$

$\displaystyle 12$

None of the other responses is correct.

$\displaystyle 15$

Correct answer:

$\displaystyle 13$

Explanation:

$\sqrt{11^{2}+ 6 \cdot 8}$ $\displaystyle \sqrt{11^{2}+ 6 \cdot 8}$

$= \sqrt{121+ 6 \cdot 8}$ $\displaystyle = \sqrt{121+ 6 \cdot 8}$

$= \sqrt{121+ 48}$ $\displaystyle = \sqrt{121+ 48}$

$= \sqrt{169}$ $\displaystyle = \sqrt{169}$

=13 $\displaystyle =13$

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Example Question #5 : How To Find The Square Root

Which of the following is equal to $\sqrt{2^{3}+ 6^{2}+10^{2}}$ $\displaystyle \sqrt{2^{3}+ 6^{2}+10^{2}}$ ?

Possible Answers:

$\displaystyle 11$

None of the other responses is correct.

$\displaystyle 14$

$\displaystyle 13$

$\displaystyle 12$

Correct answer:

$\displaystyle 12$

Explanation:

$\sqrt{2^{3}+ 6^{2}+10^{2}}$ $\displaystyle \sqrt{2^{3}+ 6^{2}+10^{2}}$

$=\sqrt{8+36 +100}$ $\displaystyle =\sqrt{8+36 +100}$

$=\sqrt{ 144}$ $\displaystyle =\sqrt{ 144}$

=12 $\displaystyle =12$

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Example Question #3 : How To Find The Square Root

$V^{2} = 72$ $\displaystyle V^{2} = 72$ ; $\displaystyle V$ is positive.

Which is the greater quantity?

(a) $\displaystyle V$

(b) $\displaystyle 8$

Possible Answers:

It is impossible to tell which is greater from the information given

(B) is greater

(A) is greater

(A) and (B) are equal

Correct answer:

(A) is greater

Explanation:

Since $\displaystyle V$ is positive, we can compare $\displaystyle V$ to 8 by comparing their squares.

$V^{2} = 72$ $\displaystyle V^{2} = 72$ and $8^{2} = 8 \times 8 = 64$ $\displaystyle 8^{2} = 8 \times 8 = 64$

$V^{2} > 8^{2}$ $\displaystyle V^{2} > 8^{2}$ , so V>8 $\displaystyle V>8$ , making (A) greater.

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Example Question #2 : Squares / Square Roots

$F^{2}= 36$ $\displaystyle F^{2}= 36$

Which is the greater quantity?

(A) $\displaystyle F$

(B) $\displaystyle 4$

Possible Answers:

(A) and (B) are equal

(B) is greater

It is impossible to tell which is greater from the information given

(A) is greater

Correct answer:

It is impossible to tell which is greater from the information given

Explanation:

If $F^{2} = 36$ $\displaystyle F^{2} = 36$ , then one of two things is true - either F = 6 $\displaystyle F = 6$ or F = -6 $\displaystyle F = -6$ .

However, 6 > 4 $\displaystyle 6 > 4$ and -6<4 $\displaystyle -6< 4$ , so it is impossibe to tell whether (A) or (B) is greater.

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Example Question #2 : Squares / Square Roots

$Q^{2} = 49$ $\displaystyle Q^{2} = 49$

Which is the greater quantity?

(A) $\displaystyle Q$

(B) $\displaystyle -8$

Possible Answers:

(A) and (B) are equal

(B) is greater

(A) is greater

It is impossible to tell which is greater from the information given

Correct answer:

(A) is greater

Explanation:

If $Q^{2} = 49$ $\displaystyle Q^{2} = 49$ , then one of two things is true - either Q=7 $\displaystyle Q=7$ or Q=- 7 $\displaystyle Q=- 7$ . Since 7 > -8 $\displaystyle 7 > -8$ and -7 > -8 $\displaystyle -7 > -8$ , Q > -8 $\displaystyle Q > -8$ either way, so (A) is greater.

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All ISEE Middle Level Quantitative Resources

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ISEE Middle Level Quantitative : How to find the square root

Study concepts, example questions & explanations for ISEE Middle Level Quantitative

All ISEE Middle Level Quantitative Resources

Example Questions

Example Question #1 : How To Find The Square Root

Example Question #2 : How To Find The Square Root

Example Question #1 : How To Find The Square Root

Example Question #2 : How To Find The Square Root

Example Question #1 : Squares / Square Roots

Example Question #6 : How To Find The Square Root

Example Question #5 : How To Find The Square Root

Example Question #3 : How To Find The Square Root

Example Question #2 : Squares / Square Roots

Example Question #2 : Squares / Square Roots

All ISEE Middle Level Quantitative Resources

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