Chapter 6.5, Problem 56PPS

To determine

To Simplify:

  $\sqrt[b]{a^{4b}}\text{ ; b is even}$

Answer to Problem 56PPS

  ${\text{a}}^4$

Explanation of Solution

Given Information:

Given expression to simplify:

  $\sqrt[b]{a^{4b}}\text{ ; b is even}$

Concept Used :

• Definition of nth root:
• Words : For any real numbers a and b , and any positive integer n , if $a^n=b$ then a is the nth root of b . That is $a=\sqrt[n]{b}$
• $\begin{array}{l}\sqrt[n]{b}\\ Here\\ b=radicand\\ n=index\\ \sqrt{}=radicalsign\end{array}$
• When you find an even root of an even power and the result is an odd power , you must use the absolute value of the result to ensure that the answer is nonnegative.
• If the index is odd, there is only one real root. Therefore, there is no principal root when index is odd , and absolute value symbol are not needed.
• Product Property of Radicals: $\begin{array}{l}\text{For any real number a and b and any integer n > 1 , }\\ \sqrt[n]{ab}=\sqrt[n]{a}\cdot \sqrt[n]{b}\text{ , if n is even and a \& b are both nonnegative }\\ \text{or if n is odd}\text{.}\end{array}$
• Quotient Property of Radicals:

  $\begin{array}{l}\text{For any real numbers a and b}\ne \text{0 and any integer}\\ \text{n >1,}\\ \sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}},if\text{ all roots are defined}\text{.}\end{array}$

• Rationalizing the denominator :

To eliminate radicals from a denominator or fractions from a radicand , use this process.

To rationalize a denominator, multiply the numerator and denominator by a quantity so that the radicand has an exact root.

• Like radical expressions : If the radicals have both the index and radicand identical , then they are like radicals.
• In order to add or subtract , the radicals must be like radicals.

Calculation:

• Given $\sqrt[b]{a^{4b}}\text{ ; b is even}$
• $\begin{array}{l}\sqrt[b]{a^{4b}}\text{ ; b is even}\\ $ ={(a^{4b})}^{\frac{1}{b}}=a^4\text{ [Since b is even and the result has no odd exponent , no need to use absolute value symbol]}${\text{=a}}^4\text{ is the principal root}\text{. }\end{array}$
• So, ${\text{a}}^4$ is the solution.