If 3^{x} = 4^{y} = 12^{z}, then z is equal to

Correct Answer is : xyx+yxy/x + yx+yxy

Correct Answer is : xyx+yxy/x + yx+yxy

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UPSC CDS I 2019 Math Paper

Question : 37 of 100

Marks: +1, -0

3^{x}

4^{y}

12^{z}

z

is equal to

Solution:

Concept:

Use a common variable to represent the equal powers, then relate the bases using multiplication.

Explanation:

Let

3^{x} = 4^{y} = 12^{z} = k

Then

3 = k^{1/x}

4 = k^{1/y}

, and

12 = k^{1/z}

Since

3 \times 4 = 12

, substitute the expressions:

k^{1/x} \times k^{1/y} = k^{1/z}

Using the law of exponents

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

, we get:

k^{1/x + 1/y} = k^{1/z}

Since bases are equal (

k

), equate the exponents:

\frac{1}{z} = \frac{1}{x} + \frac{1}{y}

Take the reciprocal of both sides:

z = \frac{1}{\frac{1}{x} + \frac{1}{y}} = \frac{xy}{x + y}

This matches option C.

Answer:

z = \frac{xy}{x + y}

(Option C).

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