Question

which of the following is equivalent to ( (5)^{\frac{2}{3}} )?
( 5^{-4} )
( 5^{4} )
( sqrt3{5^{2}} )
( sqrt3{5^{4}} )

Explanation:

Step1: Recall the exponent rule for fractional exponents

The rule for a fractional exponent \(a^{\frac{m}{n}}\) is that it can be rewritten as \(\sqrt[n]{a^{m}}\) (or equivalently \((\sqrt[n]{a})^{m}\)). Here, we have \((5)^{\frac{7}{3}}\), so applying the rule, \(n = 3\) and \(m=7\), so it should be equivalent to \(\sqrt[3]{5^{7}}\).

Step2: Analyze each option

• Option 1: \(5^{-4}\) is \( \frac{1}{5^{4}}\), which is not equivalent to \(5^{\frac{7}{3}}\).
• Option 2: \(5^{4}=5\times5\times5\times5\), while \(5^{\frac{7}{3}}\) is a different power, so not equivalent.
• Option 3: \(\sqrt[3]{5^{5}} = 5^{\frac{5}{3}}\), which is not equal to \(5^{\frac{7}{3}}\).
• Option 4: \(\sqrt[3]{5^{7}}=5^{\frac{7}{3}}\) by the fractional exponent rule, so this is the equivalent form.

Answer:

\(\boldsymbol{\sqrt[3]{5^{7}}}\) (the fourth option, assuming the last option is \(\sqrt[3]{5^{7}}\))