If x and y are positive real numbers such that \left(\frac{x^{-1}y^{\frac{2}{3}}}{\sqrt{x^… — GRE General Test

If $x$ and $y$ are positive real numbers such that $\left(\frac{x^{-1}y^{\frac{2}{3}}}{\sqrt{x^3 y^{-2}}}\right)^{-\frac{3}{4}} = x^a y^b$ , what is the value of $a + b$ ?

$\frac{5}{8}$ Answer
B
$-\frac{5}{8}$
C
$\frac{17}{8}$
D
$-\frac{1}{8}$
E
$-\frac{13}{8}$

Answer

The value of

a + b

\frac{5}{8}

Simplifying the fraction inside the parentheses gives

x^{-\frac{5}{2}} y^{\frac{5}{3}}

. Raising this to the power of

-\frac{3}{4}

produces

x^{\frac{15}{8}} y^{-\frac{5}{4}}

. Equating exponents yields

a = \frac{15}{8}

and

b = -\frac{5}{4} = -\frac{10}{8}

. Adding these together gives

a + b = \frac{5}{8}

Step-by-Step Solution

Rewrite the radical expression in the denominator using fractional exponents.

\sqrt{x^3 y^{-2}} = (x^3 y^{-2})^{\frac{1}{2}} = x^{\frac{3}{2}} y^{-1}

Applying the power rule

(u^v)^w = u^{v \cdot w}

to radical expressions converts square roots to fractional exponents of

\frac{1}{2}

Simplify the expression inside the main parentheses by combining like base exponents.

\frac{x^{-1}y^{\frac{2}{3}}}{x^{\frac{3}{2}} y^{-1}} = x^{-1 - \frac{3}{2}} y^{\frac{2}{3} - (-1)} = x^{-\frac{5}{2}} y^{\frac{5}{3}}

When dividing exponential expressions with the same base, subtract the denominator exponent from the numerator exponent.

Apply the outer exponent

-\frac{3}{4}

to each factor inside the parentheses.

(x^{-\frac{5}{2}} y^{\frac{5}{3}})^{-\frac{3}{4}} = x^{(-\frac{5}{2})(-\frac{3}{4})} y^{(\frac{5}{3})(-\frac{3}{4})} = x^{\frac{15}{8}} y^{-\frac{5}{4}}

According to exponent rules,

(u \cdot v)^p = u^p \cdot v^p

and

(u^p)^q = u^{p \cdot q}

Identify

a

and

b

and calculate their sum

a + b

a = \frac{15}{8}, \quad b = -\frac{5}{4} = -\frac{10}{8} \implies a + b = \frac{15}{8} - \frac{10}{8} = \frac{5}{8}

Match the simplified expression with

x^a y^b

and add the resulting fractional exponents using a common denominator.

Key Concept

Simplifying nested algebraic expressions with negative and rational exponents using fundamental laws of exponents.

Estimated Time:2m 0s

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