Question

simplify the radical expression.
\sqrt4{x^{19}}
write your answer in the form a, \sqrtn{b}, or a\sqrtn{b}, where k 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 for division

$x^{18} = x^{4 \times 4 + 2} = x^{16} \cdot x^2$

Step2: Split radical into product

$\sqrt[4]{x^{18}} = \sqrt[4]{x^{16} \cdot x^2} = \sqrt[4]{x^{16}} \cdot \sqrt[4]{x^2}$

Step3: Simplify 4th root of $x^{16}$

$\sqrt[4]{x^{16}} = (x^{16})^{\frac{1}{4}} = x^4$

Step4: Add absolute value for non-negativity

Since even roots require non-negative results, $\sqrt[4]{x^{16}} = |x^4|$

Step5: Combine terms

$|x^4|\sqrt[4]{x^2}$

Answer:

$|x^4|\sqrt[4]{x^2}$