Square Roots & Cube Roots Questions and Answers

Answer: A

The least number to add is 38.

\(62^{2} = 3844\) and \(63^{2} = 3969\), so 3931 lies between them, and the next perfect square is 3969.

\[ 3969 - 3931 = 38 \]

Adding 63, 69 or 169 gives 3994, 4000 and 4100, none of which is a perfect square (\(64^{2} = 4096\)).

Answer: B

The square root of 0.000169 is 0.013.

Write the number as a whole number over a power of ten with an even exponent, so the root can be taken of each part:

\[ \begin{aligned} 0.000169 &= \frac{169}{10^{6}} \\ \sqrt{0.000169} &= \frac{13}{10^{3}} = 0.013 \end{aligned} \]

Check: \(0.013 \times 0.013 = 0.000169\). The shortcut is that a square root has half as many decimal places as the number: six places become three. 0.13 squares to 0.0169, 0.0013 squares to 0.00000169 and 0.00013 is smaller still, so each misplaces the decimal point.

Answer: C

The sum is 138.75.

Each number under a root is 15625 with the decimal point moved two places, and moving the point two places under a square root moves it one place in the root:

\[ \begin{aligned} \sqrt{15625} &= 125 \\ \sqrt{156.25} &= 12.5 \\ \sqrt{1.5625} &= 1.25 \end{aligned} \]

Adding them, \(125 + 12.5 + 1.25 = 138.75\). The options 13.875 and 1.3875 are the same digits with the decimal point misplaced, and 156.25 is not a sum of the roots at all.

Answer: C

The square root is \(1\frac{2}{13}\).

Convert the mixed number to an improper fraction, then take the square roots of numerator and denominator separately.

\[ \begin{aligned} 1\tfrac{56}{169} &= \tfrac{169 + 56}{169} = \tfrac{225}{169} \\ \sqrt{\tfrac{225}{169}} &= \tfrac{15}{13} = 1\tfrac{2}{13} \end{aligned} \]

Check: \(\left(\frac{15}{13}\right)^{2} = \frac{225}{169}\). \(1\frac{1}{13} = \frac{14}{13}\) squares to \(\frac{196}{169}\), too small, and the options beginning with 2 square to more than 4, far too large.

Answer: A

The square root of 0.0169 is 0.13.

0.0169 has four decimal places, so its square root has two. Since \(13^{2} = 169\):

\[ \begin{aligned} \sqrt{0.0169} &= \sqrt{\frac{169}{10000}} \\ &= \frac{13}{100} = 0.13 \end{aligned} \]

Check: \(0.13 \times 0.13 = 0.0169\). 1.3 squared is 1.69 and 0.013 squared is 0.000169, so both misplace the decimal point.

A quick check is to count decimal places: a square root has half as many as the number itself, so four places become two.

Answer: C

Each student gives as many paise as there are students, so the total is the square of the number of students.

\[ \text{Rs. } 59.29 = 5929 \text{ paise} \]

\[ \sqrt{5929} = 77 \]

Answer: B

\(8+2\sqrt{15}\\= 5+3 + 2 \times\sqrt{5} \times \sqrt{3}\\=(\sqrt{5})^2+(\sqrt{3})^2 + (2 \times\sqrt{5} \times \sqrt{3})\\= (\sqrt{5} +\sqrt{3} )^2\\ \text{Hence, }\sqrt{\left(8+2\sqrt{15}\right)} = \sqrt{(\sqrt{5} +\sqrt{3} )^2} = \sqrt{5} +\sqrt{3}\)

Answer: C

Work outward from the inner root.

\[ \sqrt{64} = 8 \quad\Longrightarrow\quad \sqrt{248 + 8} = \sqrt{256} \]

\[ = 16 \]

Answer: C

\(\left ( \sqrt3 -\dfrac{1}{\sqrt3} \right )^2\\ =\left ( \sqrt 3 \right )^2 + \left ( \dfrac{1}{\sqrt3} \right )^2 -2 \times \sqrt 3 \times \dfrac{1}{\sqrt3}\\ = 3 + \dfrac{1}{3} - 2\\ =1 + \dfrac{1}{3}\\ =\dfrac{4}{3}\)

Answer: C

Rows times trees-per-row equals the total, and the two are equal.

\[ n^{2} = 49{,}284 \quad\Longrightarrow\quad n = \sqrt{49284} = 222 \]

Checking: \( 222^{2} = 49{,}284 \).