For positive real values of u and v, the expression \left( \frac{9u^{-1}v^2}{4u^3v^{-4}} \… — ACT

For positive real values of $u$ and $v$ , the expression $\left( \frac{9u^{-1}v^2}{4u^3v^{-4}} \right)^{-1/2}$ can be simplified to which of the following?

A
$\frac{2u^2v}{3}$
$\frac{2u^2}{3v^3}$ Answer
C
$\frac{3u^2}{2v^3}$
D
$\frac{2}{3uv^3}$
E
$\frac{4u^2}{9v^3}$

Answer

The expression is equivalent to

\frac{2u^2}{3v^3}

The correct answer is obtained by first simplifying the quotient inside the parenthesis to get

\frac{9}{4}u^{-4}v^6

. Then, raising each factor to the

-1/2

power yields

\left(\frac{9}{4}\right)^{-1/2} = \frac{2}{3}

(u^{-4})^{-1/2} = u^2

, and

(v^6)^{-1/2} = v^{-3}

. Combining these results and rewriting with positive exponents gives the simplified expression.

Step-by-Step Solution

Simplify the

u

terms inside the parenthesis using the quotient rule for exponents.

\frac{u^{-1}}{u^3} = u^{-1 - 3} = u^{-4}

The quotient rule states that

\frac{x^a}{x^b} = x^{a-b}

Simplify the

v

terms inside the parenthesis using the quotient rule for exponents.

\frac{v^2}{v^{-4}} = v^{2 - (-4)} = v^6

Subtracting a negative exponent is equivalent to adding its absolute value.

Apply the outer exponent of

-1/2

to the coefficient.

\left(\frac{9}{4}\right)^{-1/2} = \left(\frac{4}{9}\right)^{1/2} = \frac{2}{3}

A negative exponent represents taking the reciprocal of the base, and a fractional exponent of

1/2

represents the square root.

Apply the outer exponent of

-1/2

to the simplified variable terms using the power of a power rule.

(u^{-4})^{-1/2} = u^2

and

(v^6)^{-1/2} = v^{-3}

The power of a power rule states that

(x^a)^b = x^{ab}

Combine the simplified parts and rewrite the expression with positive exponents.

\frac{2}{3} u^2 v^{-3} = \frac{2u^2}{3v^3}

An expression with a negative exponent in the numerator can be moved to the denominator with a positive exponent.

Key Concept

Applying product, quotient, and power rules of exponents to algebraic expressions with negative and rational exponents.

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