Chapter EP, Problem 7.6.16EP

To determine

To find: The simplified value of the given expression.

Answer to Problem 7.6.16EP

  $\frac{3125}{32}$

Explanation of Solution

Given information:

An expression is given as ${\left(\frac{48}{1875}\right)}^{-\frac{5}{4}}$ .

Concepts used:

For any real non zero number $a$ and any integers $x$ and $y$ with $y>1$ ,

  $a^{\frac{x}{y}}={\left(\sqrt[y]{a}\right)}^x$ , except when $a<0$ and $y$ is even (in this case a complex root may exist).

Calculation:

Given expression is ${\left(\frac{48}{1875}\right)}^{-\frac{5}{4}}$ .

Simplifying -

  $\begin{array}{l}{\left(\frac{48}{1875}\right)}^{-\frac{5}{4}}\\ ={\left(\frac{16\times 3}{625\times 3}\right)}^{-\frac{5}{4}}\\ ={\left(\frac{16}{625}\right)}^{-\frac{5}{4}}\\ ={\left(\frac{625}{16}\right)}^{\frac{5}{4}}\left\{\text{Using property, }{a}^{-n}=\frac{1}{a^n}\right\}\\ =\frac{{\left(625\right)}^{\frac{5}{4}}}{{\left(16\right)}^{\frac{5}{4}}}\left\{\text{Quotient property}\right\}\\ =\frac{{\left(5\right)}^{4\times \frac{5}{4}}}{{\left(2\right)}^{4\times \frac{5}{4}}}\left\{\text{Factoring}\right\}\\ =\frac{5^5}{2^5}\\ =\frac{3125}{32}\end{array}$

Hence, simplified value of given expression is $\frac{3125}{32}$ .