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# Square Roots & Cube Roots

1.

The cube root of 0.000216 is:

$$\sqrt[3]{0.000216}\\ = \left ( \dfrac{216}{10^6} \right )^{\dfrac{1}{3}}\\ = \left ( \dfrac {6 \times 6 \times 6}{10^2 \times 10^2 \times 10^2} \right )^{\dfrac{1}{3}}\\ = \dfrac{6}{10^2}\\ = \dfrac{6}{100}\\ = 0.06$$

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2.

Evaluate: $$\sqrt{248+\sqrt{64}}$$

$$\sqrt{248+\sqrt{64}}\\ =\sqrt{248+8}\\ =\sqrt{256}\\ =16$$

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3.

Students of a class collected as many paisa from each student of class as is the number of students in that class. If total collection is Rs. 59.29, then find the total number of students in the class.

So from the question it is clear that total sum collected was 59.29 rupees.
So total paisa are 5929.
$$\text{Total Members = } \sqrt{5929} \\ = 77$$

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4.

What is the smallest number by which 3600 be divided to make it a perfect cube?

$$3600 = 2^3 \times 5^2 \times 3^2 \times 2$$
Therefore, To make it a perfect cube it must be divided by
$$5^2 \times 3^2 \times 2 = 450$$

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5.

What is the square root of $$\left(8+2\sqrt{15}\right)$$ ?

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

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6.

If $$3\sqrt{5} + \sqrt{125}= 17.88$$ , then what will be the value of$$\sqrt{80} + 16\sqrt{5}$$ ?

$$3\sqrt{5} + \sqrt{125}= 17.88\\ \Rightarrow 3\sqrt{5} + \sqrt{25 \times 5}= 17.88\\ \Rightarrow 3\sqrt{5} + 5\sqrt{5}= 17.88\\ \Rightarrow 8\sqrt{5}= 17.88\\ \Rightarrow \sqrt{5}= \dfrac{17.88}{8} = 2.235\\\\ \therefore \sqrt{80} + 16\sqrt{5}\\= \sqrt{16 \times 5} + 16\sqrt{5} \\ 4\sqrt{5} + 16\sqrt{5}\\= 20\sqrt{5}\\ = 20 \times 2.235\\ = 44.7$$

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7.

The least perfect square, which is divisible by each of 21, 36 and 66 is:

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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8.

Simplify: $$\left ( \sqrt3 -\dfrac{1}{\sqrt3} \right )^2$$

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

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9.

$$\sqrt[3]{4\dfrac{12}{125}}\text{ = ?}$$

$$\sqrt[3]{4\dfrac{12}{125}}\\=\sqrt[3]{\dfrac{512}{125}}\\ = \sqrt[3]{\dfrac{2\times 2 \times 2 \times 2\times 2 \times 2 \times 2\times 2 \times 2}{5 \times 5 \times 5}}\\ =\dfrac{2\times 2 \times 2 }{5}\\ = \dfrac{8}{5}\\ = 1\dfrac{3}{5}$$

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10.

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:

Let $$n$$ be the number of rows
Then $$n \times n = 49284$$
$$\text{ie, n = }\sqrt{49284} = 222$$

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