Question

Simplify and express each as a rational number:

(i) ${\left(\frac{4}{9}\right)}^6\times {\left(\frac{4}{9}\right)}^{-4}$
(ii) ${\left(\frac{-7}{8}\right)}^{-3}\times {\left(\frac{-7}{8}\right)}^2$
(iii) ${\left(\frac{4}{3}\right)}^{-3}\times {\left(\frac{4}{3}\right)}^{-2}$

Answer 1

(i) ${\left(\frac{4}{9}\right)}^6\times {\left(\frac{4}{9}\right)}^{-4}={\left(\frac{4}{9}\right)}^{6+\left(-4\right)}$ $\left[s\mathrm{ince}{\mathrm{a}}^{\mathrm{n}}\times {\mathrm{a}}^{\mathrm{m}}={\mathrm{a}}^{\mathrm{n}+\mathrm{m}}\right]$
= ${\left(\frac{4}{9}\right)}^2$ = $\frac{{\left(4\right)}^2}{{\left(9\right)}^2}=\frac{4\times 4}{9\times 9}=\frac{16}{81}$

(ii) ${\left(\frac{-7}{8}\right)}^{-3}\times {\left(\frac{-7}{8}\right)}^2={\left(\frac{-7}{8}\right)}^{\left(-3\right)+2}$ $\left[s\mathrm{ince}{\mathrm{a}}^{\mathrm{n}}\times {\mathrm{a}}^{\mathrm{m}}={\mathrm{a}}^{\mathrm{n}+\mathrm{m}}\right]$
= ${\left(\frac{-7}{8}\right)}^{-1}$
= ${\left(\frac{8}{-7}\right)}^1$ $\left[s\mathrm{ince}{\left(\frac{a}{b}\right)}^{-1}={\left(\frac{b}{a}\right)}^1\right]$
= $\left(\frac{8\times -1}{-7\times -1}\right)=\frac{-8}{7}$

(iii) ${\left(\frac{4}{3}\right)}^{-3}\times {\left(\frac{4}{3}\right)}^{-2}={\left(\frac{4}{3}\right)}^{\left(-3\right)+\left(-2\right)}$ $\left[s\mathrm{ince}{\mathrm{a}}^{\mathrm{n}}\times {\mathrm{a}}^{\mathrm{m}}={\mathrm{a}}^{\mathrm{n}+\mathrm{m}}\right]$
= ${\left(\frac{4}{3}\right)}^{-5}$
= ${\left(\frac{3}{4}\right)}^5$ $\left[s\mathrm{ince}{\left(\frac{a}{b}\right)}^{-1}={\left(\frac{b}{a}\right)}^1\right]$
= $\frac{{\left(3\right)}^5}{{\left(4\right)}^5}=\frac{3\times 3\times 3\times 3\times 3}{4\times 4\times 4\times 4\times 4}=\frac{243}{1024}$