Chapter 3.2, Problem 44E

Solution

To simplify the expression ${\left(y^4\right)}^{\frac{1}{6}}$ .

  $y^{\frac{2}{3}}$

Given:

The expression ${\left(y^4\right)}^{\frac{1}{6}}$ .

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: $\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}$
• 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 ${\left(y^4\right)}^{\frac{1}{6}}$ , use the exponent properties and simplify further as shown below,

  $\begin{array}{c}{\left(y^4\right)}^{\frac{1}{6}}=y^{4\cdot \frac{1}{6}}\\ =y^{\frac{4}{6}}\\ =y^{\frac{2}{3}}\end{array}$

Therefore, the simplified expression is $y^{\frac{2}{3}}$ .