Question

Simplify:

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

Answer 1

Given expression:

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

Step-wise calculation:

1. Apply the power of a power rule (a^m)ⁿ = a^mn:

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

2. Use the difference of squares formula (m + n)(m – n) = m² – n²:

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

3. Multiply terms in numerator with the same base by adding exponents:

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

4. Simplify by subtracting exponents in the numerator and denominator:

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

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

The given expression simplifies to 1.