Given that $x > 0$ , $\sqrt{x^{3}}$ is equivalent to which of the following expressions?
$I .$ $x + x^{\frac{1}{2}}$
$I I .$ ${(x^{\frac{1}{2}})}^{3}$
$I I I .$ $(x^{2}) (x^{- \frac{1}{2}})$

None

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I

and

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 C $I I$ and $I I I$ only
$\sqrt{x^{3}} = (x^{3})^{\frac{1}{2}} = x^{\frac{3}{2}}$
$I . x + x^{\frac{1}{2}}$
This cannot be further simplified since there is addition between the two terms.
$I I . (x^{\frac{1}{2}})^{3} = x^{\frac{1}{2} \times 3} = x^{\frac{3}{2}}$
$I I I . (x^{2}) (x^{- \frac{1}{2}}) = x^{(2 - \frac{1}{2})} = x^{\frac{3}{2}}$
Hence option $C$ is correct.

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