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 formula for rational exponents

The formula to convert a rational exponent \(a^{\frac{m}{n}}\) to a radical is \(\sqrt[n]{a^m}\), where \(n\) is the index of the radical and \(m\) is the exponent of the base inside the radical. Also, \((ab)^c=a^c b^c\) by the power - of - a - product rule.

For the expression \((8t)^{\frac{3}{8}}\), using the power - of - a - product rule \((ab)^c=a^c b^c\), we can rewrite it as \(8^{\frac{3}{8}}t^{\frac{3}{8}}\). But we can also directly apply the formula for converting from rational exponents to radicals. The general form \(x^{\frac{m}{n}}=\sqrt[n]{x^m}\), so for \((8t)^{\frac{3}{8}}\), we have \(\sqrt[8]{(8t)^3}\) (since \(x = 8t\), \(m = 3\), and \(n = 8\)).

Step2: Analyze the other options

• Option 2: \(8\sqrt[8]{t^3}\) would be the case if we incorrectly applied the power - of - a - product rule as \(8^{\frac{3}{8}}t^{\frac{3}{8}}=8\times8^{-\frac{5}{8}}t^{\frac{3}{8}}\), which is not equivalent to \((8t)^{\frac{3}{8}}\).
• Option 3: \(\sqrt[8]{8t^3}\) would be the case if we incorrectly applied the power - of - a - product rule as \(8^{\frac{1}{8}}t^{\frac{3}{8}}\), which is not equivalent to \((8t)^{\frac{3}{8}}\).
• Option 4: \(t^3\) is clearly not equivalent to \((8t)^{\frac{3}{8}}\) because the base is \(8t\) not \(t\) and the exponent is \(\frac{3}{8}\) not \(3\).

Answer:

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