If the algebraic expression \frac{(x^a y^2)^{-3} (x^2 y^b)^2}{(x^{-3} y^{-1})^{-2}} simpli… — ACT

If the algebraic expression $\frac{(x^a y^2)^{-3} (x^2 y^b)^2}{(x^{-3} y^{-1})^{-2}}$ simplifies to $x^4 y^4$ for all non-zero real numbers $x$ and $y$ , where $a$ and $b$ are integers, what is the value of $a + b$ ?

Answer: 4

Answer

The value of

a + b

4

Applying exponent rules to the expression

\frac{(x^a y^2)^{-3} (x^2 y^b)^2}{(x^{-3} y^{-1})^{-2}}

yields

\frac{x^{-3a}y^{-6} \cdot x^4 y^{2b}}{x^6 y^2} = \frac{x^{-3a+4} y^{2b-6}}{x^6 y^2} = x^{-3a-2} y^{2b-8}

. Equating these exponents to the target expression

x^4 y^4

gives the system

-3a - 2 = 4

and

2b - 8 = 4

. Solving these equations gives

a = -2

and

b = 6

, resulting in a sum of

a + b = 4

Step-by-Step Solution

Simplify the terms in the numerator.

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

and

(x^2 y^b)^2 = x^4 y^{2b}

Apply the power of a product rule:

(u^m v^n)^p = u^{mp} v^{np}

Multiply the simplified terms in the numerator.

x^{-3a+4} y^{2b-6}

Apply the product rule of exponents by adding exponents of like bases:

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

Simplify the denominator.

(x^{-3} y^{-1})^{-2} = x^6 y^2

Apply the power of a product rule.

Divide the numerator by the denominator.

x^{-3a-2} y^{2b-8}

Apply the quotient rule of exponents by subtracting denominator exponents from numerator exponents:

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

Set up equations by equating the simplified exponents to the exponents in the target expression

x^4 y^4

-3a - 2 = 4

and

2b - 8 = 4

For the expressions to be equivalent for all non-zero real numbers, the corresponding exponents of

x

and

y

must be equal.

Solve the linear equations for the integer constants

a

and

b

a = -2

and

b = 6

Isolate the variables:

-3a = 6 \implies a = -2

, and

2b = 12 \implies b = 6

Find the sum of

a

and

b

4

Add the values of the constants:

-2 + 6 = 4

Key Concept

Properties of exponents (product, quotient, and power rules) in multi-step algebraic simplification

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