Question

which expression is equivalent to ((8t)^{\frac{3}{8}})? (sqrt8{(8t)^3}) (8sqrt8{t^3}) (sqrt8{8t^3}) (t^3)

Explanation:

Step1: Recall the exponent - radical conversion rule

The rule for converting a rational exponent to a radical is \(a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\), where \(a\) is the base, \(m\) is the numerator of the exponent, and \(n\) is the denominator of the exponent.

For the expression \((8t)^{\frac{3}{8}}\), using the rule \(a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\) with \(a = 8t\), \(m=3\), and \(n = 8\), we get \(\sqrt[8]{(8t)^{3}}\).

Step2: Analyze the other options

• Option 2: \(8\sqrt[8]{t^{3}}\) would be the case if we incorrectly applied the exponent rule by splitting \(8t\) as \(8\times t\) and only applying the exponent to \(t\), which is wrong. The exponent \(\frac{3}{8}\) applies to the entire base \(8t\), not just \(t\).
• Option 3: \(\sqrt[8]{8t^{3}}\) is incorrect because we need to raise the entire base \(8t\) to the power of 3, not just \(t\).
• Option 4: \(t^{3}\) is incorrect as it ignores the base \(8\) and the radical form conversion.

Answer:

\(\boldsymbol{\sqrt[8]{(8t)^{3}}}\) (the first option)