For all non-zero real numbers x and y, the expression \frac{(x^2 y^{-3})^{-2} (x^{-1} y^4)… — ACT

For all non-zero real numbers $x$ and $y$ , the expression

\frac{(x^2 y^{-3})^{-2} (x^{-1} y^4)^3}{(x^3 y^{-2})^d}

can be written in the form $x^p y^q$ , where $p$ and $q$ are integers. If $q = 2p$ , what is the value of $d$ ?

Answer: -4

-4

Applying the rules of exponents yields the simplified expression

x^{-7-3d} y^{18+2d}

. Setting the exponent of

y

equal to twice the exponent of

x

gives the equation

18+2d = 2(-7-3d)

, which solves to

d = -4

Apply the power of a power rule to the terms in the numerator.

(x^2 y^{-3})^{-2} = x^{-4} y^6

and

(x^{-1} y^4)^3 = x^{-3} y^{12}

To raise a power to another power, multiply the exponents:

(u^m)^n = u^{mn}

Multiply the simplified terms in the numerator together.

x^{-4} y^6 \cdot x^{-3} y^{12} = x^{-7} y^{18}

To multiply powers with the same base, add the exponents:

u^m \cdot u^n = u^{m+n}

Apply the power of a power rule to the denominator.

(x^3 y^{-2})^d = x^{3d} y^{-2d}

Distribute the exponent

d

to both variables inside the parentheses by multiplying the exponents.

Divide the numerator by the denominator.

\frac{x^{-7} y^{18}}{x^{3d} y^{-2d}} = x^{-7-3d} y^{18-(-2d)} = x^{-7-3d} y^{18+2d}

To divide powers with the same base, subtract the exponent of the denominator from the exponent of the numerator:

\frac{u^m}{u^n} = u^{m-n}

Set up the linear equation for

d

using

q = 2p

and solve.

18+2d = 2(-7-3d) \implies 18+2d = -14-6d \implies 8d = -32 \implies d = -4

The problem states the relationship between the final exponents is

q = 2p

, where

p = -7-3d

and

q = 18+2d

Properties of exponents (product, quotient, and power rules) combined with solving a linear equation.

Question