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What is the value of the expression, $$\left(\frac{1}{x^{a-b}}\right)^{\frac{1}{a-c}} \left(\frac{1}{x^{b-c}}\right)^{\frac{1}{b-a}} \left(\frac{1}{x^{c-a}}\right)^{\frac{1}{c-b}}\quad ?$$

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Are you familiar with the basic properties of exponents? $1/q=q^{-1}$, $(q^r)^s=q^{rs}$, $q^uq^v=q^{u+v}$? –  Gerry Myerson Jan 27 '13 at 4:11
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What have you tried, and where did you get stuck? –  anorton Jan 27 '13 at 4:12
    
Are we allowed to take more that 20 minutes this time? –  Michael Albanese Jan 27 '13 at 11:15

1 Answer 1

$$\biggl(\frac{1}{x^{a−b}}\biggr)^{^\frac{1}{a−c}}=\biggl({x^{b-a}}\biggr)^{^\frac{1}{a−c}}$$

$$=\biggl({x}\biggr)^{^\frac{b-a}{a−c}}$$

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

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