What is the smallest number by which 3600 be divided to make it a perfect cube?
Answer: A
Factorise and see which primes have exponents that are not multiples of 3.
\[ 3600 = 2^{4} \times 3^{2} \times 5^{2} \]
Dividing by \( 2 \times 3^{2} \times 5^{2} = 450 \) leaves
\[ \frac{3600}{450} = 8 = 2^{3} \]
a perfect cube. So the smallest divisor is 450.
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What is the square root of \(\left(8+2\sqrt{15}\right)\) ?
Answer: B
\(8+2\sqrt{15}\\= 5+3 + 2 \times\sqrt{5} \times \sqrt{3}\\=(\sqrt{5})^2+(\sqrt{3})^2 + (2 \times\sqrt{5} \times \sqrt{3})\\= (\sqrt{5} +\sqrt{3} )^2\\ \text{Hence, }\sqrt{\left(8+2\sqrt{15}\right)} = \sqrt{(\sqrt{5} +\sqrt{3} )^2} = \sqrt{5} +\sqrt{3}\)
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If \(3\sqrt{5} + \sqrt{125}= 17.88\) , then what will be the value of\(\sqrt{80} + 16\sqrt{5}\) ?
Answer: D
\(3\sqrt{5} + \sqrt{125}= 17.88\\ \Rightarrow 3\sqrt{5} + \sqrt{25 \times 5}= 17.88\\ \Rightarrow 3\sqrt{5} + 5\sqrt{5}= 17.88\\ \Rightarrow 8\sqrt{5}= 17.88\\ \Rightarrow \sqrt{5}= \dfrac{17.88}{8} = 2.235\\\\ \therefore \sqrt{80} + 16\sqrt{5}\\= \sqrt{16 \times 5} + 16\sqrt{5} \\ 4\sqrt{5} + 16\sqrt{5}\\= 20\sqrt{5}\\ = 20 \times 2.235\\ = 44.7\)
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The least perfect square, which is divisible by each of 21, 36 and 66 is:
Answer: A
L.C.M. of \(21, 36, 66 = 2772\)
Now, \(2772 = 2 \times 2 \times 3 \times 3 \times 7 \times 11\)
To make it a perfect square, it must be multiplied by \(7 \times 11\)
So, required number
\(= 22 \times 32 \times 72 \times 112\\
= 213444\)
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Simplify: \(\left ( \sqrt3 -\dfrac{1}{\sqrt3} \right )^2\)
Answer: C
\(\left ( \sqrt3 -\dfrac{1}{\sqrt3} \right )^2\\ =\left ( \sqrt 3 \right )^2 + \left ( \dfrac{1}{\sqrt3} \right )^2 -2 \times \sqrt 3 \times \dfrac{1}{\sqrt3}\\ = 3 + \dfrac{1}{3} - 2\\ =1 + \dfrac{1}{3}\\ =\dfrac{4}{3}\)
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\(\sqrt[3]{4\dfrac{12}{125}}\text{ = ?}\)
Answer: C
\(\sqrt[3]{4\dfrac{12}{125}}\\=\sqrt[3]{\dfrac{512}{125}}\\ = \sqrt[3]{\dfrac{2\times 2 \times 2 \times 2\times 2 \times 2 \times 2\times 2 \times 2}{5 \times 5 \times 5}}\\ =\dfrac{2\times 2 \times 2 }{5}\\ = \dfrac{8}{5}\\ = 1\dfrac{3}{5}\)
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A man plants 49284 apple trees in his garden and arranges them so that there are as many rows as there are apples trees in each row. The number of rows is:
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: \( 222^{2} = 49{,}284 \).
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Evaluate: \(\sqrt[3]{\sqrt{.000064}}\)
Answer: D
\(\sqrt{.000064} \\ = \sqrt{\dfrac{64}{10^6}}\\ =\dfrac{8}{10^3}\\ = .008\)
Now,
\(\sqrt[3]{.008}\\ =\sqrt[3]{\dfrac{8}{1000}}\\ = \dfrac{2}{10}\\ = 0.2\)
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