If a and b represent positive real numbers, which of the following is an equivalent form o… — ACT

If $a$ and $b$ represent positive real numbers, which of the following is an equivalent form of the expression below?

\frac{(a^{-1} + b^{-1})^{-2}}{ab}

The correct answer is the fraction with

ab

in the numerator and the square of the sum

(a+b)

in the denominator, which is \frac{ab}{(a+b)^2}.

The correct answer is found by first rewriting

a^{-1} + b^{-1}

\frac{1}{a} + \frac{1}{b}

, which simplifies to

\frac{a+b}{ab}

using a common denominator. Raising this to the power of

-2

gives

\frac{a^2 b^2}{(a+b)^2}

. Finally, dividing this expression by

ab

reduces the exponents of

a

and

b

by 1, resulting in

\frac{ab}{(a+b)^2}

Rewrite the negative exponents in the numerator as reciprocals.

The expression inside the parentheses becomes

\frac{1}{a} + \frac{1}{b}

By the definition of negative exponents,

x^{-1} = \frac{1}{x}

Find a common denominator to add the fractions inside the parentheses.

\frac{1}{a} + \frac{1}{b} = \frac{b}{ab} + \frac{a}{ab} = \frac{a+b}{ab}

To add fractions, they must share a common denominator, which is the product of

a

and

b

Apply the negative power of

-2

to the simplified fraction.

\left(\frac{a+b}{ab}\right)^{-2} = \left(\frac{ab}{a+b}\right)^2 = \frac{a^2 b^2}{(a+b)^2}

An expression raised to a negative exponent is equal to the reciprocal of the expression raised to the positive exponent.

Divide the result by the denominator

ab

\frac{a^2 b^2}{(a+b)^2} \cdot \frac{1}{ab} = \frac{ab}{(a+b)^2}

Dividing by a term is equivalent to multiplying by its reciprocal, and applying the exponent quotient rule simplifies

\frac{a^2 b^2}{ab}

ab

Properties of exponents including negative power rules, quotient rules, and algebraic fraction addition.

An alternative method is to substitute small positive integer values for

a

and

b

. Let

a = 1

and

b = 2

. The expression evaluates to

\frac{(1^{-1} + 2^{-1})^{-2}}{1 \cdot 2} = \frac{(1 + 0.5)^{-2}}{2} = \frac{1.5^{-2}}{2} = \frac{(3/2)^{-2}}{2} = \frac{4/9}{2} = \frac{2}{9}

. Evaluating the correct expression with these values yields

\frac{1 \cdot 2}{(1+2)^2} = \frac{2}{9}

, which matches.

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