Question

select the expression that is equivalent to \\(\frac{1}{(5x)^{-\frac{5}{4}}}\\)

answer
\\(\circ\\) \\(\frac{1}{\sqrt5{(5x)^4}}\\) \\(\circ\\) \\(\frac{1}{\sqrt4{(5x)^5}}\\)
\\(\circ\\) \\(\sqrt5{(5x)^4}\\) \\(\circ\\) \\(\sqrt4{(5x)^5}\\)

Explanation:

Step1: Recall negative exponent rule

The negative exponent rule states that \(a^{-n}=\frac{1}{a^{n}}\), so \(\frac{1}{a^{-n}} = a^{n}\). Applying this to the denominator \((5x)^{-\frac{5}{4}}\), we get \(\frac{1}{(5x)^{-\frac{5}{4}}}=(5x)^{\frac{5}{4}}\).

Step2: Convert exponent to radical form

Recall that \(a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\). For \((5x)^{\frac{5}{4}}\), \(n = 4\) and \(m=5\), so \((5x)^{\frac{5}{4}}=\sqrt[4]{(5x)^{5}}\).

Answer:

\(\boldsymbol{\sqrt[4]{(5x)^{5}}}\) (corresponding to the option \(\boldsymbol{\sqrt[4]{(5x)^{5}}}\))