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UPSC CDS II 2025 Mathematics Paper

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Question : 22 of 100

Marks: +1, -0

If a^{b} = b^{a}, then what is

\frac{a\times (\frac{a}{b})^{\frac{a}{b}}}{a^{(\frac{a}{b})}}

equal to?

1
ab
b
a^{b}

Solution:

Concept:

Use exponent rules and the given relation

a^b = b^a

to simplify the expression.

Explanation:

Let

E = \frac{a \times (\frac{a}{b})^{\frac{a}{b}}}{a^{\frac{a}{b}}}

.

Combine the powers:

E = a \times \left(\frac{a/b}{a}\right)^{\frac{a}{b}} = a \times \left(\frac{1}{b}\right)^{\frac{a}{b}} = a \times b^{-\frac{a}{b}}

.

From

a^b = b^a

, take both sides to the power

\frac{1}{ab}

:

a^{1/a} = b^{1/b}

.

Raise both sides to the power

b

:

b^{a/b} = a

. Hence

b^{-a/b} = \frac{1}{a}

.

Substitute into

E

:

E = a \times \frac{1}{a} = 1

.

Answer:

1

(Option A)

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