Chapter 3, Problem 13T

Solution

To simplify the expression $\sqrt[3]{64x^4{y}^2}$ .

  $4x\sqrt[3]{x{y}^2}$

Given:

The expression $\sqrt[3]{64x^4{y}^2}$ .

Concept Used:

• Power of a power property: ${\left(x^a\right)}^b=x^{a\cdot b}$
• Product of power property: $a^m\cdot a^n=a^{m+n}$
• Quotient of power property: $\frac{a^m}{a^n}=a^{m-n}$
• Product property of radicals: $\sqrt[n]{a}\cdot \sqrt[n]{b}=\sqrt[n]{a\cdot b}$
• Quotient property of radicals: $\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}$
• Sum of like radicals’ property: $p\sqrt[n]{a}+q\sqrt[n]{a}=\left(p+q\right)\sqrt[n]{a}$
• The exponent form of the nth root of a number a is given by $\sqrt[n]{a}=a^{\frac{1}{n}}$

Calculation:

In order to simplify the given expression $\sqrt[3]{64x^4{y}^2}$ , use the exponent properties and simplify further as shown below,

  $\begin{array}{c}\sqrt[3]{64x^4{y}^2}=\sqrt[3]{64}\cdot \sqrt[3]{x^4}\cdot \sqrt[3]{y^2}\\ =\sqrt[3]{4^3}\cdot \sqrt[3]{x^3\cdot x}\cdot \sqrt[3]{y^2}\\ =4\cdot \sqrt[3]{x^3}\cdot \sqrt[3]{x}\cdot \sqrt[3]{y^2}\\ =4\cdot x\cdot \sqrt[3]{x}\cdot \sqrt[3]{y^2}\\ =4\cdot x\cdot \sqrt[3]{x\cdot y^2}\\ =4x\sqrt[3]{x{y}^2}\end{array}$

Therefore, the simplified expression is $4x\sqrt[3]{x{y}^2}$ .