Questions: Which expression is equivalent to (4^7/3) ? (7 sqrt[3]4) ((sqrt[7]4)^3) ((sqrt[3]4)^7) (3 sqrt[7]4)

Solution Steps

To solve the problem of converting the expression \(4^{\frac{7}{3}}\) into an equivalent radical form, we need to understand the relationship between rational exponents and radicals. The expression \(a^{\frac{m}{n}}\) can be rewritten as \((\sqrt[n]{a})^m\) or \(\sqrt[n]{a^m}\). Therefore, \(4^{\frac{7}{3}}\) can be expressed as \((\sqrt[3]{4})^7\).

The given expression is \(4^{\frac{7}{3}}\). We need to convert this expression into an equivalent radical form.

The expression \(a^{\frac{m}{n}}\) can be rewritten as \((\sqrt[n]{a})^m\). Applying this to our expression, we have: \[ 4^{\frac{7}{3}} = (\sqrt[3]{4})^7 \]

To verify the equivalence, we calculate \((\sqrt[3]{4})^7\). The cube root of 4 is approximately \(1.5874\), and raising this to the power of 7 gives approximately \(25.3984\).

Final Answer