Question

prove the above given in the picture
please help me to solve and please tell it step by step​

Answer 1

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)