Chapter 3, Problem 3T

Solution

To evaluate the expression $\sqrt[4]{81}$ without using a calculator.

  $3$

Given:

The expression $\sqrt[4]{81}$ .

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 evaluate the given expression $\sqrt[4]{81}$ , use the exponent properties and simplify further as shown below,

  $\begin{array}{c}\sqrt[4]{81}=\sqrt[4]{3^4}\\ =3\end{array}$

Therefore, the value of the expression is $3$ .