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Prove That: `1/(1+X^(B-a)+X^(C-a))+1/(1+X^(A-b)+X^(C-b))+1/(1+X^(B-c)+X^(A-c))=1` - Mathematics

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Prove that:

`1/(1+x^(b-a)+x^(c-a))+1/(1+x^(a-b)+x^(c-b))+1/(1+x^(b-c)+x^(a-c))=1`

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Solution

Consider the left hand side:

`1/(1+x^(b-a)+x^(c-a))+1/(1+x^(a-b)+x^(c-b))+1/(1+x^(b-c)+x^(a-c))`

`=1/(1+x^b/x^a+x^c/x^a)+1/(1+x^a/x^b+x^c/x^b)+1/(1+x^b/x^c+x^a/x^c)`

`=1/((x^a+x^b+x^c)/x^a)+1/((x^b+x^a+x^c)/x^b)+1/((x^c+x^b+x^a)/x^c)`

`=x^a/(x^a+x^b+x^c)+x^b/(x^b+x^a+x^c)+x^c/(x^c+x^b+x^a)`

`=(x^a+x^b+x^c)/(x^a+x^b+x^c)`

= 1

Therefore left hand side is equal to the right hand side. Hence proved.

Concept: Laws of Exponents for Real Numbers
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APPEARS IN

RD Sharma Mathematics for Class 9
Chapter 2 Exponents of Real Numbers
Exercise 2.1 | Q 4.2 | Page 12

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