Question

which expression has the same meaning as (5^{\frac{4}{3}})? options: (sqrt3{8^4}), (sqrt4{8^3}), (sqrt3{5^4}), (sqrt4{5^3})

Explanation:

Step1: Recall the exponent rule

The rule for fractional exponents is \( a^{\frac{m}{n}}=\sqrt[n]{a^{m}} \). For \( 5^{\frac{3}{4}} \), \( a = 5 \), \( m=3 \), \( n = 4 \).

Step2: Apply the rule

Using the rule \( a^{\frac{m}{n}}=\sqrt[n]{a^{m}} \), we substitute \( a = 5 \), \( m = 3 \), \( n=4 \). So \( 5^{\frac{3}{4}}=\sqrt[4]{5^{3}} \).

Answer:

\(\boldsymbol{\sqrt[4]{5^{3}}}\) (the option with \(\sqrt[4]{5^{3}}\))