Question

simplify the radical expression,
\sqrt{x^{5}}

write your answer in the form a, \sqrt{b}, or a\sqrt{b}, where a and b are constants or expressions in x. use at most one radical in your answer, and at most one absolute value symbol in your expression for a.

Explanation:

Step1: Rewrite exponent as sum

$x^5 = x^{4+1} = x^4 \cdot x$

Step2: Split radical into product

$\sqrt{x^5} = \sqrt{x^4 \cdot x} = \sqrt{x^4} \cdot \sqrt{x}$

Step3: Simplify $\sqrt{x^4}$

$\sqrt{x^4} = |x^2| = x^2$ (since $x^2$ is non-negative for all real $x$)

Step4: Combine terms

$x^2 \cdot \sqrt{x}$

Answer:

$x^2\sqrt{x}$