Prove that- i 1-1-xa-b-1-1-xb-a-1 ii 1-1-xb-a-xc-a-1-1-xa-b-xc-b-1-1-xb-c-xa-c-1

(i) 1/1+x^a b+1/1+x^b a=1 LHS=1/1+x^a b+1/1+x^b a =1/x^b b+x^a b+1/x^a a+x^b a =1/x^ b(x^b+x^a)+1/x^ a(x^a+x^b) =x^b/x^a+x^b+x^a/x^a+x^b =x^b+x^a/(x^a+x^b)=x^a+ ...

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Byju's Answer

Standard IX

Mathematics

Laws of Exponents for Real Numbers

Prove that: i...

Question

Prove that:

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

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

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Solution

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

$L H S = \frac{1}{1 + x^{a - b}} + \frac{1}{1 + x^{b - a}} = \frac{1}{x^{b - b} + x^{a - b}} + \frac{1}{x^{a - a} + x^{b - a}} = \frac{1}{x^{- b} (x^{b} + x^{a})} + \frac{1}{x^{- a} (x^{a} + x^{b})} = \frac{x^{b}}{x^{a} + x^{b}} + \frac{x^{a}}{x^{a} + x^{b}} = \frac{x^{b} + x^{a}}{(x^{a} + x^{b})} = \frac{x^{a} + x^{b}}{(x^{a} + x^{b})} = 1 = R H S$

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

$L H S = \frac{1}{1 + x^{b - a} + x^{c - a}} + \frac{1}{1 + x^{a - b} + x^{c - b}} + \frac{1}{1 + x^{b - c} + x^{a - c}} = \frac{1}{x^{a - a} + x^{b - a} + x^{c - a}} + \frac{1}{x^{b - b} + x^{a - b} + x^{c - b}} + \frac{1}{x^{c - c} + x^{b - c} + x^{a - c}} {P u t 1 = x^{a - a}, 1 = x^{b - b} a n d 1 = x^{c - c}} = \frac{1}{x^{- a} (x^{a} + x^{b} + x^{c})} + \frac{1}{x^{- b} (x^{b} + x^{a} + x^{c})} + \frac{1}{x^{- c} (x^{c} + x^{b} + x^{a})} = \frac{x^{a}}{x^{a} + x^{b} + x^{c}} + \frac{x^{b}}{x^{a} + x^{b} + x^{c}} + \frac{x^{c}}{x^{a} + x^{b} + x^{c}} = \frac{x^{a} + x^{b} + x^{c}}{x^{a} + x^{b} + x^{c}} = 1 = R H S$

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Similar questions

Q. Prove that:

(i)

\frac{1}{1 + x^{a - b}} + \frac{1}{1 + x^{b - a}} = 1

(ii)

\frac{1}{1 + x^{b - a} + x^{c - a}} + \frac{1}{1 + x^{a - b} + x^{c - b}} + \frac{1}{1 + x^{b - c} + x^{a - c}} = 1

Q. Prove that

\frac{1}{1 + x^{b - a} + x^{c - a}} + \frac{1}{1 + x^{a - b} + x^{c - b}} + \frac{1}{1 + x^{b - c} + x^{a - c}} = 1

Q. Prove that :

\frac{1}{1 + x^{b - a} + x^{c - a}} + \frac{1}{1 + x^{a - b} + x^{c - b}} + \frac{1}{1 + x^{b - c} + x^{a - c}} = 1

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

Q. Simplify :

\frac{1}{1 + x^{b - a} + x^{c - a}} + \frac{1}{1 + x^{a - b} + x^{c - b}} + \frac{1}{1 + x^{a - c} + x^{b - c}}

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Prove that: (i) 11+xa−b+11+xb−a=1 (ii) 11+xb−a+xc−a+11+xa−b+xc−b+11+xb−c+xa−c=1

Prove that:

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

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