If x is a positive real number such that x^{\frac{3}{4}} = 27, what is the value of x^{-\f… — GRE General Test

If $x$ is a positive real number such that $x^{\frac{3}{4}} = 27$ , what is the value of $x^{-\frac{1}{2}}$ ?

\frac{1}{9}

Raising both sides of

x^{\frac{3}{4}} = 27

to the power of

\frac{4}{3}

gives

x = (3^3)^{\frac{4}{3}} = 3^4 = 81

. Substituting

x = 81

into

x^{-\frac{1}{2}}

gives

81^{-\frac{1}{2}} = \frac{1}{\sqrt{81}} = \frac{1}{9}

Solve for

x

in the equation

x^{\frac{3}{4}} = 27

x = 27^{\frac{4}{3}} = (3^3)^{\frac{4}{3}} = 3^4 = 81

Raise both sides to the power of

\frac{4}{3}

to isolate

x

Evaluate

x^{-\frac{1}{2}}

for

x = 81

81^{-\frac{1}{2}} = \frac{1}{81^{\frac{1}{2}}} = \frac{1}{\sqrt{81}} = \frac{1}{9}

Apply the negative exponent rule

a^{-n} = \frac{1}{a^n}

and the fractional exponent rule

a^{\frac{1}{2}} = \sqrt{a}

Fractional and Negative Exponents

Estimated Time:1m 30s

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