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UPSC CDS 2 2020 Math Paper

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Question : 13 of 100

Marks: +1, -0

If

\mathbf{x}^{\mathbf{m}}=\sqrt[14]{\mathbf{x} \sqrt{\mathbf{x} \sqrt{\mathbf{x}}}}

, then what is the value of m?

$\frac{1}{8}$
$\frac{1}{4}$
$\frac{3}{4}$
$\frac{7}{4}$

Solution:

Concept:

Use the rules of exponents:

\sqrt[n]{a} = a^{1/n}

and

a^m \cdot a^n = a^{m+n}

.

Explanation:

Start with the given equation:

x^m = \sqrt[14]{x \sqrt{x \sqrt{x}}}

.

Convert the innermost square root:

\sqrt{x} = x^{1/2}

.

Then

x \sqrt{x} = x \cdot x^{1/2} = x^{3/2}

.

The next square root:

\sqrt{x \sqrt{x}} = \sqrt{x^{3/2}} = (x^{3/2})^{1/2} = x^{3/4}

.

Now the expression inside the 14th root becomes:

x \cdot \sqrt{x \sqrt{x}} = x \cdot x^{3/4} = x^{1 + 3/4} = x^{7/4}

.

Take the 14th root:

\sqrt[14]{x^{7/4}} = (x^{7/4})^{1/14} = x^{(7/4) \cdot (1/14)} = x^{7/56} = x^{1/8}

.

Thus

x^m = x^{1/8}

, so

m = \frac{1}{8}

.

Answer:

m = \frac{1}{8}

(Option A).

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