Question

Simplify:

(i) ${\left[{\left\{{\left(\frac{-1}{4}\right)}^2\right\}}^{-2}\right]}^{-1}$
(ii) ${\left\{{\left(\frac{-2}{3}\right)}^2\right\}}^3$
(iii) ${\left(\frac{-3}{2}\right)}^3÷{\left(\frac{-3}{2}\right)}^6$
(iv) ${\left(\frac{-2}{3}\right)}^7÷{\left(\frac{-2}{3}\right)}^4$

Answer 1

(i)

${\left[{\left\{{\left(-\frac{1}{4}\right)}^2\right\}}^{-2}\right]}^{-1}={\left[{\left(-\frac{1}{4}\right)}^{2\times -2}\right]}^{-1}$ $\left[s\mathrm{ince}{\left\{{\left(\frac{\mathrm{a}}{\mathrm{b}}\right)}^{\mathrm{m}}\right\}}^{\mathrm{n}}={\left(\frac{\mathrm{a}}{\mathrm{b}}\right)}^{\mathrm{mn}}\right]$
$={\left[{\left(-\frac{1}{4}\right)}^{-4}\right]}^{-1}$
$$\begin{gathered} ={\left(-\frac{1}{4}\right)}^{\left(-4\right)\times \left(-1\right)} \\ ={\left(-\frac{1}{4}\right)}^4=\frac{{\left(-1\right)}^4}{{\left(4\right)}^4} \\ =\frac{1}{256} \end{gathered}$$

(ii)

${\left\{{\left(\frac{-2}{3}\right)}^2\right\}}^3={\left(\frac{-2}{3}\right)}^{2\times 3}$ $\left[s\mathrm{ince}{\left\{{\left(\frac{\mathrm{a}}{\mathrm{b}}\right)}^{\mathrm{m}}\right\}}^{\mathrm{n}}={\left(\frac{\mathrm{a}}{\mathrm{b}}\right)}^{\mathrm{mn}}\right]$
= ${\left(\frac{-2}{3}\right)}^6$
= $\frac{{\left(-2\right)}^6}{{\left(3\right)}^6}=\frac{64}{729}$ [since (−2)⁶ = 64 and (3)⁶ = 729]

(iii)

${\left(\frac{-3}{2}\right)}^3÷{\left(\frac{-3}{2}\right)}^6={\left(\frac{-3}{2}\right)}^{3-6}$ $\left[s\mathrm{ince}{\mathrm{a}}^{\mathrm{m}}÷{\mathrm{a}}^{\mathrm{n}}={\mathrm{a}}^{\mathrm{m}-\mathrm{n}}\right]$
= ${\left(\frac{-3}{2}\right)}^{-3}$
$={\left(\frac{2}{-3}\right)}^3$ $\left[s\mathrm{ince}{\left(\frac{\mathrm{a}}{\mathrm{b}}\right)}^{-1}={\left(\frac{\mathrm{b}}{\mathrm{a}}\right)}^1\right]$
$$\begin{gathered} ={\left(\frac{2\times -1}{-3\times -1}\right)}^3={\left(\frac{-2}{3}\right)}^3 \\ =\frac{{\left(-2\right)}^3}{{\left(3\right)}^3}=\frac{-8}{27} \end{gathered}$$

(iv)

${\left(\frac{-2}{3}\right)}^7÷{\left(\frac{-2}{3}\right)}^4={\left(\frac{-2}{3}\right)}^{7-4}$ $\left[s\mathrm{ince}{\mathrm{a}}^{\mathrm{m}}÷{\mathrm{a}}^{\mathrm{n}}={\mathrm{a}}^{\mathrm{m}-\mathrm{n}}\right]$
$={\left(\frac{-2}{3}\right)}^3$
= $\frac{{\left(-2\right)}^3}{{\left(3\right)}^3}=\frac{-8}{27}$