Question

`(a^x/(a^-y))^(x - y) (a^y/a^-z)^(y - z) (a^z/a^-x)^(z - x)` is equal to ______.

Options

• 0

• 1

• a

• a^xyz

Answer 1

`(a^x/(a^-y))^(x - y) (a^y/a^-z)^(y - z) (a^z/a^-x)^(z - x)` is equal to 1.

Explanation:

We are given the expression:

`(a^x/(a^-y))^(x - y) xx (a^y/a^-z)^(y - z) xx (a^z/a^-x)^(z - x)`

Step 1: Simplify each fraction using exponent rules:

`a^x/(a^-y) = a^(x + y)`

`a^y/a^-z = a^(y + z)`

`a^z/a^-x = a^(z + x)`

Step 2: Plug back into the expression:

`(a^(x + y))^(x - y) xx(a^(y + z))^(y - z) xx (a^(z + x))^(z - x)`

Use the rule (a^m)ⁿ = a^{m × n}:

`a^((x + y)(x - y)) xx a^((y + z)(y - z)) xx a^((z + x)(z - x))`

Step 3: Expand each exponent using identity (a + b)(a – b) = a² – b²:

`a^(x^2 - y^2) xx a^(y^2 - z^2) xx a^(z^2 - x^2)`

Now combine the powers:

`a^((x^2 - y^2 + y^2 - z^2 + z^2 - x^2)) = a^0`

`a^((x^2 - y^2 + y^2 - z^2 + z^2 - x^2)) = 1`