prove the above given in the picture
please help me to solve and please tell it step by step
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Step-by-step explanation:
Given :-
[x^a/x^b]^(a²+ab+b²)×[x^b/x^c]^(b²+bc+c²)
×[x^c/x^a]^(c²+ca+a²)
To find :-
Prove that :
[x^a/x^b]^(a²+ab+b²)×[x^b/x^c]^(b²+bc+c²)
×[x^c/x^a]^(c²+ca+a²) = 1
Solution:-
On taking LHS
[x^a/x^b]^(a²+ab+b²)×[x^b/x^c]^(b²+bc+c²)
×[x^c/x^a]^(c²+ca+a²)
=> [x^(a-b)]^(a²+ab+b²)×[x^(b-c)]^(b²+bc+c²)
×[x^(c-a)]^(c²+ca+a²)
Since a^m / a^n = a^(m-n)
=> [x^(a-b)(a²+ab+b²)]×[x^(b-c)(b²+bc+c²)]
×[x^(c-a)(c²+ca+a²)]
Since (a^m)^n = a^mn
We know that a³-b³ = (a-b)(a²+ab+b²)
=> [x^(a³-b³)]×[x^(b³-c³)]×[x^(c³-a³)]
=> x^[(a³-b³)+(b³-c³)+(c³-a³)]
Since a^m × a^n = a^(m+n)
=> x^[a³-b³+b³-c³+c³-a³]
=>x^0
=> 1
Since a^0 = 1
=>RHS
=>LHS = RHS
Verified the given realtion in the given problem
Used formulae:-
- a^m × a^n = a^(m+n)
- a^m / a^n = a^(m-n)
- (a^m)^n = a^mn
- a^0 = 1
- a³-b³ = (a-b)(a²+ab+b²)
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