If `(z^(x ^(2 ) +y ^(2)))/(z ^(-2xy ))= (z ^(3)),` x and y are positive integers, and `x gt y,` what is the value of `x -y` ?

Options

A. 1
B. 2
C. 3
D. 8

Answer

A

Explanation

Use exponent rules to simplify the fraction. Since the terms in the numerator and denominator have the same base, z, subtract the exponent of the denominator from that of the numerator: `z ^(x ^(2)+y ^(2)-(-2xy))=z^(x ^(2)+2xy + y^(2)).` When a value with an exponents. So, for the term on the right side, `(z ^(3))^(3)=z ^(9).` Now the equation base, z their exponents must be equal and you can write `x ^(2) + 2xy + y^(2) =9.` Factor `x ^(2)+ 2xy + y^(2),` a classic quadratic, to get `(x +y) ^(2) =9.` Thus, `x +y` must be either 3 or -3. However, the question limits x and y to positive integers, so they must be 1 and 2. Furthermore, since `x gt y,` x must be 2 and y must be 1. Consequently, `x -y =1-1 =1.` (A) is correct.