Question

select the expression that is equivalent to \\(\frac{1}{2x^{-\frac{2}{5}}}\\)
answer
\\(\sqrt{2x^5}\\) \\(\sqrt5{2x^2}\\) \\(\frac{\sqrt5{x^2}}{2}\\) \\(\frac{\sqrt{x^5}}{2}\\)

Explanation:

Step1: Recall negative exponent rule

The negative exponent rule states that \( a^{-n}=\frac{1}{a^{n}} \), so we can rewrite \( x^{-\frac{2}{5}} \) as \( \frac{1}{x^{\frac{2}{5}}} \). Then the original expression \( \frac{1}{2x^{-\frac{2}{5}}} \) becomes \( \frac{1}{2\cdot\frac{1}{x^{\frac{2}{5}}}} \).

Step2: Simplify the fraction

Simplifying \( \frac{1}{2\cdot\frac{1}{x^{\frac{2}{5}}}} \), we multiply the numerator and denominator by \( x^{\frac{2}{5}} \), getting \( \frac{x^{\frac{2}{5}}}{2} \).

Step3: Recall fractional exponent to radical form

The fractional exponent rule states that \( a^{\frac{m}{n}}=\sqrt[n]{a^{m}} \). So \( x^{\frac{2}{5}}=\sqrt[5]{x^{2}} \). Substituting this back, we have \( \frac{\sqrt[5]{x^{2}}}{2} \).

Answer:

\(\frac{\sqrt[5]{x^{2}}}{2}\) (the option corresponding to this expression)