Question

Simplify : xpower b /x power c whole power b+c-a into xpower c /x power a whole power c+a-b into xpower a /x power b whole power a+b-c

Answer 1

Question:

Simplify the expression given in the question

$(\frac{x^b}{x^c})^{b+c-a} *(\frac{x^c}{x^a})^{c+a-b} *(\frac{x^a}{x^b})^{a+b-c}$

Answer:

$(\frac{x^b}{x^c})^{b+c-a} *(\frac{x^c}{x^a})^{c+a-b} *(\frac{x^a}{x^b})^{a+b-c}$ = 1

Given:

$(\frac{x^b}{x^c})^{b+c-a} *(\frac{x^c}{x^a})^{c+a-b} *(\frac{x^a}{x^b})^{a+b-c}$

To Find:

Simplify $(\frac{x^b}{x^c})^{b+c-a} *(\frac{x^c}{x^a})^{c+a-b} *(\frac{x^a}{x^b})^{a+b-c}$

Solution:

• $x^a * x^b = x^{a+b}$
• $\frac{x^a}{x^b}= x^{a-b}$

• a⁰ = 1

$(\frac{x^b}{x^c})^{b+c-a} *(\frac{x^c}{x^a})^{c+a-b} *(\frac{x^a}{x^b})^{a+b-c}$

= $(x^{b-c})^{b+c-a} *(x^{c-a})^{c+a-b} *(x^{a-b})^{a+b-c}$

= $(x^{b^2+bc-ab-bc-c^2+ac}) *(x^{c^2+ac-bc-ac-a^2+ab}) *(x^{a^2+ab-ac-ab-b^2+bc})$

= $(x^{b^2-ab-c^2+ac}) *(x^{c^2-bc-a^2+ab}) *(x^{a^2-ac-b^2+bc})$

= $x^{b^2-ab-c^2+ac +c^2-bc-a^2+ab +a^2-ac-b^2+bc}$

= x⁰

= 1

Therefore, $(\frac{x^b}{x^c})^{b+c-a} *(\frac{x^c}{x^a})^{c+a-b} *(\frac{x^a}{x^b})^{a+b-c}$ = 1

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