Chapter 3, Problem 13REAE

Solution

To evaluate the expression ${27}^{-\frac{2}{3}}$ without using a calculator.

  $\frac{1}{9}$

Given:

The expression ${27}^{-\frac{2}{3}}$ .

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 ${27}^{-\frac{2}{3}}$ , use the exponent properties and simplify further as shown below,

  $\begin{array}{c}{27}^{-\frac{2}{3}}={\left(3^3\right)}^{-\frac{2}{3}}\\ =3^{3\cdot \left(-\frac{2}{3}\right)}\\ =3^{-2}\\ =\frac{1}{3^2}\\ =\frac{1}{9}\end{array}$

Therefore, the value of the expression is $\frac{1}{9}$ .