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# Prove that : ((X^A)/(X^B))^( a + B - C ) (( X^B)/(X^C))^( B + C - a )((X^C)/(X^A))^( C + a - B) - Mathematics

Sum

Prove that : ((x^a)/(x^b))^( a + b - c ) (( x^b)/(x^c))^( b + c - a )((x^c)/(x^a))^( c + a - b)

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#### Solution

LHS = ((x^a)/(x^b))^( a + b - c ) (( x^b)/(x^c))^( b + c - a )((x^c)/(x^a))^( c + a - b)

= (x^(a - b))^( a + b - c) xx (x^(b - c))^( b + c - a ) xx ( x^(c - a ))^( c + a - b)

= x^[( a - b )( a + b - c )] xx x^[( b - c )( b + c - a )] xx x^[( c - a )(c + a - b)]

= x^( a^2 + ab - ac - ab - b^2 + bc) xx x^(b^2 + bc - ab - cd - c^2 + ac) xx x^( c^2 + ac - bc - ac - a^2 + ab )

= x^(a^2 - ac - b^2 + bc + b^2 - ab - c^2 + ac + c^2 - bc - a^2 + ab )
= x0
= 1
= RHS

Concept: Solving Exponential Equations
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#### APPEARS IN

Selina Concise Mathematics Class 9 ICSE
Chapter 7 Indices (Exponents)
Exercise 7 (B) | Q 7.1 | Page 100
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