Question

1. rewrite the expressions as equivalent radical expressions.
2. $2^{\frac{3}{4}}=\square$
3. $2^{\frac{4}{3}}=\square$
4. $2^{\frac{1}{4}}=\square$

options: $\sqrt4{2^3}$, $\sqrt3{4^2}$, $\sqrt4{2}$, $\sqrt3{2^4}$

Explanation:

Step1: Recall the exponent - radical rule

The rule for converting an exponential expression \(a^{\frac{m}{n}}\) to a radical expression is \(a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\), where \(n\) is the index of the radical and \(m\) is the power of the base inside the radical.

Step2: Solve for \(2^{\frac{3}{4}}\)

For the expression \(2^{\frac{3}{4}}\), using the rule \(a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\) with \(a = 2\), \(m=3\) and \(n = 4\), we get \(\sqrt[4]{2^{3}}\).

Step3: Solve for \(2^{\frac{4}{3}}\)

For the expression \(2^{\frac{4}{3}}\), using the rule \(a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\) with \(a=2\), \(m = 4\) and \(n=3\), we get \(\sqrt[3]{2^{4}}\). Also, note that \(2^{\frac{4}{3}}=(2^{2})^{\frac{2}{3}} = 4^{\frac{2}{3}}\), and using the rule again, \(4^{\frac{2}{3}}=\sqrt[3]{4^{2}}\).

Step4: Solve for \(2^{\frac{1}{4}}\)

For the expression \(2^{\frac{1}{4}}\), using the rule \(a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\) with \(a = 2\), \(m = 1\) and \(n=4\), we get \(\sqrt[4]{2}\).

Answer:

1. \(2^{\frac{3}{4}}=\boldsymbol{\sqrt[4]{2^{3}}}\)
2. \(2^{\frac{4}{3}}=\boldsymbol{\sqrt[3]{4^{2}}}\) (or \(\boldsymbol{\sqrt[3]{2^{4}}}\))
3. \(2^{\frac{1}{4}}=\boldsymbol{\sqrt[4]{2}}\)