Chapter 7, Problem 50CR

To determine

Tosimplify:The given expression by writing all steps.

Answer to Problem 50CR

The simplified expression is ${\left(\frac{5a^8}{10a^6}\right)}^{-3}=\frac{8}{a^6}$ .

Explanation of Solution

Given information:

The given expression is ${\left(\frac{5a^8}{10a^6}\right)}^{-3}$

Formulaused:

The rules ${\left(a\right)}^{-m}={\left(\frac{1}{a}\right)}^m$ , $\frac{a^m}{a^n}=a^{m-n}$ , $\frac{a^m}{a^n}=a^{m-n}$ , ${\left(ab\right)}^m=a^m\cdot b^m$ , ${\left(a^m\right)}^n=a^{mn}$

Calculations:

The given expression is $E={\left(\frac{5a^8}{10a^6}\right)}^{-3}$

  $\Rightarrow E={\left(\frac{5}{10}\cdot \frac{a^8}{a^6}\right)}^{-3}$

  $\Rightarrow E={\left(\frac{1}{2}\cdot a^{8-6}\right)}^{-3}$$\because \frac{a^m}{a^n}=a^{m-n}$ , ${\left(a^m\right)}^n=a^{mn}$ .

  $\Rightarrow E={\left(\frac{1}{2}\right)}^{-3}\cdot {\left(a^2\right)}^{-3}$$\because {\left(ab\right)}^m=a^m\cdot b^m$

  $\Rightarrow E={\left(2\right)}^3\cdot \left(a^{2\times -3}\right)$$\because {\left(a\right)}^{-m}={\left(\frac{1}{a}\right)}^m$ , ${\left(a^m\right)}^n=a^{mn}$

  $\Rightarrow E={\left(2\right)}^3\cdot \left(a^{-6}\right)$

  $\Rightarrow E=8\cdot \frac{1}{a^6}=\frac{8}{a^6}$

Thus, the simplified expression is ${\left(\frac{5a^8}{10a^6}\right)}^{-3}=\frac{8}{a^6}$ .

Conclusion:

The simplified expression is ${\left(\frac{5a^8}{10a^6}\right)}^{-3}=\frac{8}{a^6}$ .