Chapter 3, Problem 6T

Solution

To evaluate the expression ${16}^{-\frac{3}{2}}$ without using a calculator.

  $\frac{1}{64}$

Given:

The expression ${16}^{-\frac{3}{2}}$ .

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 evaluate the given expression ${16}^{-\frac{3}{2}}$ , use the exponent properties and simplify further as shown below,

  $\begin{array}{c}{16}^{-\frac{3}{2}}=\frac{1}{{16}^{\frac{3}{2}}}\\ =\frac{1}{{\left(4^2\right)}^{\frac{3}{2}}}\\ =\frac{1}{4^{2\cdot \frac{3}{2}}}\\ =\frac{1}{4^3}\\ =\frac{1}{64}\end{array}$

Therefore, the value of the expression is $\frac{1}{64}$ .