Question

(4^{\frac{1}{2}})^2 = 4\
let x = 4^{\frac{1}{2}}\
x^2 = 4\
x = \sqrt{4}\
use the two different equations for x above to express 4^{\frac{1}{2}} as a radical in the answer form below.\
4^{\frac{1}{2}} = \square

Explanation:

Step1: Recall the definition of rational exponents

We know that \( a^{\frac{1}{n}}=\sqrt[n]{a} \) for any positive real number \( a \) and positive integer \( n \). In the case where \( a = 4 \) and \( n=2 \), we can apply this rule.

Step2: Substitute the values into the formula

Given \( x = 4^{\frac{1}{2}} \) and \( x=\sqrt{4} \) from the equations above, by the definition of rational exponents (or by equating the two expressions for \( x \)), we can say that \( 4^{\frac{1}{2}}=\sqrt{4} \). Also, we know that \( \sqrt{4} = 2 \), but the question asks to express \( 4^{\frac{1}{2}} \) as a radical, so we use the radical form.

Answer:

\(\sqrt{4}\) (or if we simplify the radical, it is \( 2 \), but since the question asks for the radical form, \(\sqrt{4}\) is the radical expression for \( 4^{\frac{1}{2}} \))