Chapter 5.CT, Problem 1CT

To determine

To evaluate:

The given numerical expression ${\text{(4)}}^{-\frac{\text{5}}{\text{2}}}$.

Answer to Problem 1CT

Solution:

${\text{(4)}}^{-\frac{\text{5}}{\text{2}}}=\frac{\text{1}}{\text{32}}.$

Explanation of Solution

If $\frac{m}{n}$ is a rational number, where n is a negative integer greater than 1, and b is a real number such that $\sqrt[n]{b}$ exists, then

$b^{-\frac{m}{n}}=\frac{1}{\sqrt[n]{b^m}}={\left(\frac{1}{\sqrt[n]{b}}\right)}^m$

Product Rule for Radicals:

If $\sqrt{\text{a}}$ and $\sqrt{\text{b}}$ are real numbers, then

$\sqrt{\text{a}}\text{·}\sqrt{\text{b}}\text{=}\sqrt{\text{a·b}}$

from the given numerical expression,

$m=\text{5}$

$n=-\text{2}$

$b=\text{4}$

Hence

${\text{4}}^{-\frac{\text{5}}{\text{2}}}\text{=}\frac{\text{1}}{\sqrt[\text{2}]{{\text{4}}^5}}={\left(\frac{1}{\sqrt[2]{\text{4}}}\right)}^{\text{5}}$

To simplify the denominator, express 4 into the product of two same integers. i.e.,

Then,

$\sqrt[2]{\text{4}}=\sqrt[\text{2}]{\text{2}\text{·}\text{2}}$

So,

$\frac{1}{\sqrt[2]{\text{4}}}=\frac{\text{1}}{\sqrt[\text{2}]{\text{2}\text{·}\text{2}}}$

Using the product rule,

$\sqrt[\text{2}]{\text{2}\text{·}\text{2}}=\sqrt[\text{2}]{\text{2}} \text{· }\sqrt[\text{2}]{\text{2}}$

Since ${\left(\sqrt[\text{2}]{\text{a }}\right)}^{\text{2}}=\text{a}$

$\sqrt[\text{2}]{\text{2}} \text{· }\sqrt[\text{2}]{\text{2}}={\left(\sqrt[\text{2}]{\text{2}}\right)}^{\text{2}}=\text{2}$

Now,

$\frac{\text{1}}{\sqrt[\text{2}]{\text{4}}}\text{=}\frac{\text{1}}{\text{2}}$.

Hence,

${\text{4}}^{-\frac{\text{5}}{\text{2}}}\text{ = }\frac{\text{1}}{\sqrt[\text{2}]{{\text{4}}^5}}={\left(\frac{\text{1}}{\sqrt[\text{2}]{\text{4}}}\right)}^{\text{5}}={\left(\frac{\text{1}}{\text{2}}\right)}^{\text{5}}$

${\text{4}}^{-\frac{\text{5}}{\text{2}}}=\left(\frac{\text{1}}{\text{2}}\right)· \left(\frac{\text{1}}{\text{2}}\right)·\left(\frac{\text{1}}{\text{2}}\right)·\left(\frac{\text{1}}{\text{2}}\right)·\left(\frac{\text{1}}{\text{2}}\right)$

${\text{4}}^{-\frac{\text{5}}{\text{2}}}=\frac{\text{1}}{\text{32}}$

Final statement:

The evaluation of the given numerical expression ${\text{4}}^{-\frac{\text{5}}{\text{2}}}$ gives $\frac{\text{1}}{\text{32}}$