What is the square root of $5$ to the power of $\frac{4}{3}?$

Full solution

Q. What is the square root of

5

to the power of

\frac{4}{3}?

Understand the expression: Understand the expression. $\newline$ The expression is $(\sqrt{5})^{\frac{4}{3}}$ . This means we need to take the square root of $5$ and then raise it to the power of $\frac{4}{3}$ .
Simplify square root of $5$ : Simplify the square root of $5$ . $\newline$ The square root of $5$ is an irrational number, and it cannot be simplified further. So, we keep it as $\sqrt{5}$ .
Apply power of $\frac{4}{3}$ : Apply the power of $\frac{4}{3}$ to the square root of $5$ . The power of $\frac{4}{3}$ can be broken down into two steps: squaring $(\sqrt{5})^2$ and then taking the cube root of the result to the power of $\frac{1}{3}$ . So, we first square $\sqrt{5}$ . $(\sqrt{5})^2 = 5$
Apply cube root: Apply the cube root to the result of Step $3$ . $\newline$ Now we take the cube root of $5$ to the power of $1/3$ . $\newline$ $5^{1/3}$ is the cube root of $5$ .
Raise to power of $4$ : Raise the result of Step $4$ to the power of $4$ . $\newline$ Now we raise the cube root of $5$ to the power of $4$ . $\newline$ $(5^{1/3})^4 = 5^{4/3}$
Simplify the expression: Simplify the expression. $\newline$ Since we have already established that $5^{\frac{4}{3}}$ is the final expression, we can conclude that this is the simplest form of the expression, as $5$ cannot be simplified further.