Advertisements
Advertisements
Question
Prove that:
`(x^a/x^b)^(a^2+ab+b^2)xx(x^b/x^c)^(b^2+bc+c^2)xx(x^c/x^a)^(c^2+ca+a^2)=1`
Advertisements
Solution
Consider the left hand side:
`(x^a/x^b)^(a^2+ab+b^2)xx(x^b/x^c)^(b^2+bc+c^2)xx(x^c/x^a)^(c^2+ca+a^2)`
`=x^(a(a^2+ab+b^2))/x^(b(a^2+ab+b^2))xxx^(b(b^2+bc+c^2))/x^(c(b^2+bc+c^2))xxx^(c(c^2+ca+a^2))/x^(a(c^2+ca+a^2))`
`=x^(a(a^2+ab+b^2)-b(a^2+ab+b^2))xxx^(b(b^2+bc+c^2)-c(b^2+bc+c^2))xxx^(c(c^2+ca+a^2)-a(c^2+ca+a^2))`
`=x^((a-b)(a^2+ab+b^2))xxx^((b-c)(b^2+bc+c^2))xxx^((c-a)(c^2+ca+a^2))`
`=x^((a^3-b^3))xxx^((b^3-c^3))xxx^((c^3-a^3))`
`=x^((a^3-b^3+b^3-c^3+c^3-a^3))`
`=x^0`
= 1
LHS = RHS
Hence proved.
APPEARS IN
RELATED QUESTIONS
If `a=xy^(p-1), b=xy^(q-1)` and `c=xy^(r-1),` prove that `a^(q-r)b^(r-p)c^(p-q)=1`
Simplify:
`(16^(-1/5))^(5/2)`
If `5^(3x)=125` and `10^y=0.001,` find x and y.
Solve the following equation:
`8^(x+1)=16^(y+2)` and, `(1/2)^(3+x)=(1/4)^(3y)`
Write \[\left( 625 \right)^{- 1/4}\] in decimal form.
If (x − 1)3 = 8, What is the value of (x + 1)2 ?
The value of x − yx-y when x = 2 and y = −2 is
If 9x+2 = 240 + 9x, then x =
If \[\frac{5 - \sqrt{3}}{2 + \sqrt{3}} = x + y\sqrt{3}\] , then
Find:-
`16^(3/4)`
