Question

4* rewrite the expressions as equivalent radical expressions.

1. $2^{\frac{3}{4}}=\square$
2. $2^{\frac{4}{3}}=\square$
3. $2^{\frac{1}{4}}=\square$

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

Explanation:

Step1: Recall exponent to radical rule

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

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

Using the rule \( a^{\frac{m}{n}}=\sqrt[n]{a^m} \), for \( a = 2 \), \( m = 3 \), \( n = 4 \), we get \( \sqrt[4]{2^3} \).

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

First, rewrite \( 2^{\frac{4}{3}} \) as \( 2^{1+\frac{1}{3}}=2\times2^{\frac{1}{3}} \), but also using the rule \( a^{\frac{m}{n}}=\sqrt[n]{a^m} \), \( 2^{\frac{4}{3}}=\sqrt[3]{2^4}=\sqrt[3]{16}=\sqrt[3]{8\times2}=\sqrt[3]{2^3\times2}=2\sqrt[3]{2} \)? Wait, no, wait the options have \( \sqrt[3]{4^2} \), since \( 2^4=(2^2)^2 = 4^2 \), so \( \sqrt[3]{2^4}=\sqrt[3]{4^2} \).

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

Using the rule \( a^{\frac{m}{n}}=\sqrt[n]{a^m} \), for \( a = 2 \), \( m = 1 \), \( n = 4 \), we get \( \sqrt[4]{2^1}=\sqrt[4]{2} \). Wait, but wait the third option's radical is \( \sqrt[4]{2} \)? Wait the third expression is \( 2^{\frac{1}{4}} \), so by the rule \( a^{\frac{1}{n}}=\sqrt[n]{a} \), so \( 2^{\frac{1}{4}}=\sqrt[4]{2} \).

Wait, let's re - check:

1. For \( 2^{\frac{3}{4}} \):

By \( a^{\frac{m}{n}}=\sqrt[n]{a^m} \), \( 2^{\frac{3}{4}}=\sqrt[4]{2^3} \), so matches the first radical option \( \sqrt[4]{2^3} \).

1. For \( 2^{\frac{4}{3}} \):

\( 2^{\frac{4}{3}}=\sqrt[3]{2^4}=\sqrt[3]{(2^2)^2}=\sqrt[3]{4^2} \), so matches the second radical option \( \sqrt[3]{4^2} \).

1. For \( 2^{\frac{1}{4}} \):

\( 2^{\frac{1}{4}}=\sqrt[4]{2^1}=\sqrt[4]{2} \), so matches the third radical option \( \sqrt[4]{2} \).

Wait, but the fourth radical option is \( \sqrt[3]{2^4} \), which is the same as \( \sqrt[3]{4^2} \) since \( 2^4 = 16=4^2 \).

Answer:

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