Question

Express each of the following rational numbers with a positive exponent:
(i) ${\left(\frac{3}{4}\right)}^{-2}$
(ii) ${\left(\frac{5}{4}\right)}^{-3}$
(iii) $4^3\times 4^{-9}$
(iv) ${\left\{{\left(\frac{4}{3}\right)}^{-3}\right\}}^{-4}$
(v) ${\left\{{\left(\frac{3}{2}\right)}^4\right\}}^{-2}$

Answer 1

$$\begin{gathered} \left(i\right){\left(\frac{3}{4}\right)}^{-2} \\ ={\left({\displaystyle \frac{4}{3}}\right)}^2 \end{gathered}$$ ---> (a⁻¹ = 1/a)

$$\begin{gathered} \left(ii\right){\left(\frac{5}{4}\right)}^{-3} \\ ={\left(\frac{4}{5}\right)}^3 \end{gathered}$$ ---> (a⁻¹ = 1/a)

$$\begin{gathered} \left(iii\right)4^3\times 4^{-9} \\ =4^{\left(3-9\right)}=4^{-6} \\ ={\left({\displaystyle \frac{1}{4}}\right)}^6 \end{gathered}$$ ---> (a^m x aⁿ = a^m+n)

$$\begin{gathered} \left(iv\right){\left\{{\left(\frac{4}{3}\right)}^{-3}\right\}}^{-4} \\ ={\left(\frac{4}{3}\right)}^{-4\times -3} \\ ={\left(\frac{4}{3}\right)}^{12} \end{gathered}$$ ---> ((a^m)ⁿ = a^mn)

$$\begin{gathered} \left(v\right){\left\{{\left(\frac{3}{2}\right)}^4\right\}}^{-2} \\ ={\left(\frac{3}{2}\right)}^{4\times -2} \\ ={\left(\frac{3}{2}\right)}^{-8} \\ ={\left({\displaystyle \frac{2}{3}}\right)}^8 \end{gathered}$$ ---> ((a^m)ⁿ = a^mn)