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Question : 35 of 86

Marks: +1, -0

If

a+b+c=0

Then the expression

X^{a^{2} b^{-1} c^{-1}} X^{a^{-1} b^{2} c^{-1}} X^{a^{-1} b^{-1} c^{2}}

simplifies to

$x$
$x^{2}$
$x^{3}$
$x^{4}$ .

Solution:

Concept:

The product of powers with the same base is equal to the base raised to the sum of the exponents.

Explanation:

Given expression:

X^{a^{2} b^{-1} c^{-1}} \cdot X^{a^{-1} b^{2} c^{-1}} \cdot X^{a^{-1} b^{-1} c^{2}}

.

Since base

X

is common, add exponents:

\displaystyle \frac{a^{2}}{bc} + \frac{b^{2}}{ac} + \frac{c^{2}}{ab} = \frac{a^{3}+b^{3}+c^{3}}{abc}

.

Given

a+b+c=0

, we have

a^{3}+b^{3}+c^{3}=3abc

.

Thus exponent

= \frac{3abc}{abc} = 3

.

Hence the expression simplifies to

X^{3}

.

Answer:

X^{3}

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