For all non-zero real numbers x and y, the algebraic expression \left(\frac{x^{a/2} y^{-1}… — ACT

For all non-zero real numbers $x$ and $y$ , the algebraic expression $\left(\frac{x^{a/2} y^{-1}}{x^{-2/3} y^{b/3}}\right)^{-6}$ is equivalent to $\frac{y^{12}}{x^{16}}$ , where $a$ and $b$ are integers. What is the value of $a + b$ ?

A
-14
B
1
7Answer
D
9
E
11

Answer

The correct answer is 7. Simplifying the expression inside the parentheses using the quotient rule yields

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

. Raising this expression to the power of

-6

results in

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

. Equating this to

x^{-16} y^{12}

gives

a = 4

and

b = 3

, so

a + b = 7

Step-by-Step Solution

Simplify the expression inside the parentheses using the quotient rule for exponents,

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

x^{\frac{a}{2} - \left(-\frac{2}{3}\right)} y^{-1 - \frac{b}{3}} = x^{\frac{a}{2} + \frac{2}{3}} y^{-1 - \frac{b}{3}}

To combine the base

x

and base

y

terms inside the parentheses before applying the outer power.

Apply the outer exponent of

-6

to the simplified expression using the power of a power rule,

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

x^{-6\left(\frac{a}{2} + \frac{2}{3}\right)} y^{-6\left(-1 - \frac{b}{3}\right)} = x^{-3a - 4} y^{6 + 2b}

To distribute the negative power of

-6

to each factor in the product.

Equate the resulting exponents to the exponents of the given equivalent expression,

\frac{y^{12}}{x^{16}} = x^{-16} y^{12}

, and solve the equations for

a

and

b

For

x

-3a - 4 = -16 \implies -3a = -12 \implies a = 4

. For

y

6 + 2b = 12 \implies 2b = 6 \implies b = 3

Since the expressions are equivalent for all non-zero values of

x

and

y

, their respective exponents must be equal.

Calculate the sum of

a

and

b

a + b = 4 + 3 = 7

To find the final requested value.

Key Concept

Properties of exponents in algebraic expressions including the quotient rule, power rule, and operations with fractional/negative exponents.

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