Simplifying Radicals Solutions

• To simplify a radical, rewrite the expression with    rational    exponents.

• The general form when rewriting a radical with rational exponents is: $\sqrt[d]{x^n}=x^{\frac{n}{d}}$
• The    index    of a radical expression determines the value of the denominator of the rational exponent.
  •    Square    root = Index 2
  •    Cubed    root = Index 3

Example 1

Simplify. Write the final answer in radical form.

$\sqrt{24x^{3}y}$

$$\begin{gathered} 24=2^3·3 \\ \sqrt{2^3·3^1·x^3·y^1} \end{gathered}$$

$$\begin{gathered} 2^{\frac{3}{2}}·3^{\frac{1}{2}}·x^{\frac{3}{2}}·y^{\frac{1}{2}} \\ 2x\sqrt{2·3xy} \\ 2x\sqrt{6xy} \end{gathered}$$

Example 2

Simplify. Write the final answer in radical form. 

$\sqrt[3]{24x^{10}}$

$$\begin{gathered} \sqrt[3]{2^3·3·x^{10}} \\ {2}^{\frac{3}{3}}·3^{\frac{1}{3}}·x^{\frac{10}{3}} \\ 2x^3 \sqrt[3]{3x} \end{gathered}$$