Question

Simplify.

(i) $5^4\times 5^3$ (ii) ${\left(\frac{2}{3}\right)}^6÷{\left(\frac{2}{3}\right)}^9$ (iii) ${\left(\frac{7}{2}\right)}^8\times {\left(\frac{7}{2}\right)}^{-6}$ (iv) ${\left(\frac{4}{5}\right)}^2÷\left(\frac{5}{4}\right)$

Answer 1

(i)

$$\begin{gathered} 5^4\times 5^3 \\ =5^7\left[\because a^m\times a^n=a^{m+n}\right] \end{gathered}$$

(ii)

$$\begin{gathered} {\left(\frac{2}{3}\right)}^6÷{\left(\frac{2}{3}\right)}^9 \\ ={\left(\frac{2}{3}\right)}^{6-9}\left[\because a^m÷a^n=a^{m-n}\right] \\ ={\left(\frac{2}{3}\right)}^{-3} \\ ={\left(\frac{3}{2}\right)}^3\left[\because a^{-m}=\frac{1}{a^m}\right] \end{gathered}$$

(iii)

$$\begin{gathered} {\left(\frac{7}{2}\right)}^8\times {\left(\frac{7}{2}\right)}^{-6} \\ ={\left(\frac{7}{2}\right)}^{8+\left(-6\right)}\left[\because a^m\times a^n=a^{m+n}\right] \\ ={\left(\frac{7}{2}\right)}^2 \end{gathered}$$


(iv)

$$\begin{gathered} {\left(\frac{4}{5}\right)}^2÷\left(\frac{5}{4}\right) \\ ={\left(\frac{4}{5}\right)}^2\times \left(\frac{4}{5}\right) \\ ={\left(\frac{4}{5}\right)}^{2+1}\left[\because a^m\times a^n=a^{m+n}\right] \\ ={\left(\frac{4}{5}\right)}^3 \end{gathered}$$