Chapter A.1, Problem 67E

Solution

Simplify the radical expression.

  $x\left(\frac{3^{\frac{1}{4}}{x}^{\frac{1}{2}}{y}^{\frac{1}{2}}}{2^{\frac{3}{4}}}\right)$ .

Given:

The given expression is $\sqrt[4]{\frac{3x^8{y}^2}{8x^2}}$ .

Concept Used:

• Find the largest factor in the radicand that is a perfect power of the index. Rewrite the radicand as a product of two factors, using that factor.
• Use the product rule to rewrite the radical as the product of two radicals.
• Simplify the root of the perfect power.

Calculation:

The given expression is $\sqrt[4]{\frac{3x^8{y}^2}{8x^2}}$ .

  $\begin{array}{c}\sqrt[4]{\frac{3x^8{y}^2}{8x^2}}=\sqrt[4]{\frac{3(x^8\cdot x^{-2})y^2}{8}}\\ =\sqrt[4]{\frac{3(x^8{}^{-2})y^2}{8}}\\ =\sqrt[4]{\frac{3x^6{y}^2}{8}}\\ =\sqrt[4]{\frac{3x^4{x}^2{y}^2}{2^3}}\\ =x\sqrt[4]{\frac{3x^2{y}^2}{2^3}}\\ =x{\left(\frac{3x^2{y}^2}{2^3}\right)}^{1/4}\\ =x\left(\frac{3^{\frac{1}{4}}{x}^{\frac{2}{4}}{y}^{\frac{2}{4}}}{2^{\frac{3}{4}}}\right)\\ =x\left(\frac{3^{\frac{1}{4}}{x}^{\frac{1}{2}}{y}^{\frac{1}{2}}}{2^{\frac{3}{4}}}\right)\end{array}$

Thus after simplifying $\sqrt[4]{\frac{3x^8{y}^2}{8x^2}}$ we get $x\left(\frac{3^{\frac{1}{4}}{x}^{\frac{1}{2}}{y}^{\frac{1}{2}}}{2^{\frac{3}{4}}}\right)$ .