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
Left hand side is equal to right hand side.
Hence proved.
APPEARS IN
RELATED QUESTIONS
Simplify the following
`(2x^-2y^3)^3`
Assuming that x, y, z are positive real numbers, simplify the following:
`(x^((-2)/3)y^((-1)/2))^2`
Solve the following equation:
`3^(x-1)xx5^(2y-3)=225`
Solve the following equation:
`4^(x-1)xx(0.5)^(3-2x)=(1/8)^x`
If a and b are distinct primes such that `root3 (a^6b^-4)=a^xb^(2y),` find x and y.
If `2^x xx3^yxx5^z=2160,` find x, y and z. Hence, compute the value of `3^x xx2^-yxx5^-z.`
If `x = a^(m + n), y = a^(n + l)` and `z = a^(l + m),` prove that `x^my^nz^l = x^ny^lz^m`
Which one of the following is not equal to \[\left( \frac{100}{9} \right)^{- 3/2}\]?
When simplified \[(256) {}^{- ( 4^{- 3/2} )}\] is
If x = \[\frac{2}{3 + \sqrt{7}}\],then (x−3)2 =
