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Summation by Sigma Method

If x and y ar...

Question

If x and y are integers such that $x^{2} = 64$ and $y^{3} = 64$ , which of the following could be true?
I. $x = 8$
II. $y = - 4$
III. $x + y = - 4$

I

only

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I I

only

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I

and

I I I

only

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I I

and

I I I

only

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I, I I

and

I I I

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Solution

The correct option is E $I$ and $I I I$ only
Given
$x^{2}$ $=$ $64$ and $y^{3}$ $=$ $64$

Possible values of $x$ ,
$x^{2}$ $=$ $64$
$x$ $=$ $\sqrt{64}$
$x$ $=$ $\sqrt{8^{2}}$ $o r$ $x$ $=$ $\sqrt{{(- 8)}^{2}}$
$x$ $=$ $8$ $o r$ $x$ $=$ $- 8$

Possible values of $y$ ,
$y^{3}$ $=$ $64$
$y$ $=$ $\sqrt[3]{64}$
$y$ $=$ $\sqrt[3]{4^{3}}$
$y$ $=$ $4$

$I$ . $x$ $=$ $8$ is True
$∵ x$ $=$ $8$ or $x$ $=$ $- 8$

$I I$ . $y$ $=$ $- 4$ is False
$∵ y$ $=$ $4$

$I I I$ . $x$ $+$ $y$ $=$ $- 4$ is True
When $x$ $=$ $8$ ,
$x$ $+$ $y$ $=$ $8$ $+$ $4$ $=$ $12$
When $x$ $=$ $- 8$ ,
$x$ $+$ $y$ $=$ $- 8$ $+$ $4$ $=$ $- 4$

Therefore, $I$ and $I I I$ only are True.

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