The expression ≤ft({x^{a}}{x^{-b}})^{a²-ab+b²} × ≤ft({x^{b}}{x^{-c}})^{b²-bc+c²} x ≤ft({x^{c}}{x^{-a}})^{c²-ac+a²} reduces to :

Correct Answer is : x2(a3+b3+c3)x²(a³+b³+c³)}x2(a3+b3+c3)

Correct Answer is : x2(a3+b3+c3)x²(a³+b³+c³)}x2(a3+b3+c3)

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Question : 40 of 86

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The expression

\left(\frac{x^{a}}{x^{-b}}\right)^{a^{2}-ab+b^{2}} \times \left(\frac{x^{b}}{x^{-c}}\right)^{b^{2}-bc+c^{2}} x \left(\frac{x^{c}}{x^{-a}}\right)^{c^{2}-ac+a^{2}}

Solution:

Concept:

Simplify using exponent laws and sum of cubes identity.

Explanation:

First, simplify each inner fraction:

\frac{x^{a}}{x^{-b}} = x^{a - (-b)} = x^{a+b}

Similarly,

\frac{x^{b}}{x^{-c}} = x^{b+c}

and

\frac{x^{c}}{x^{-a}} = x^{c+a}

Now the product becomes:

(x^{a+b})^{a^{2}-ab+b^{2}} \cdot (x^{b+c})^{b^{2}-bc+c^{2}} \cdot (x^{c+a})^{c^{2}-ac+a^{2}}

Apply

(x^m)^n = x^{mn}

: exponent from first term is

(a+b)(a^2 - ab + b^2) = a^3 + b^3

(sum of cubes identity).

Second term exponent:

(b+c)(b^2 - bc + c^2) = b^3 + c^3

Third term exponent:

(c+a)(c^2 - ac + a^2) = c^3 + a^3

Add all exponents:

(a^3+b^3) + (b^3+c^3) + (c^3+a^3) = 2a^3 + 2b^3 + 2c^3 = 2(a^3+b^3+c^3)

Hence expression simplifies to

x^{2(a^3+b^3+c^3)}

Answer:

Option C:

x^{2(a^{3}+b^{3}+c^{3})}

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