Question

Simplify the following :

${\left(\frac{-3}{4}\right)}^3\times \left[\frac{16}{27}-{\left(\frac{2}{3}\right)}^3\right]$.

Answer 1

Step 1: Find the value of ${\left(\frac{-3}{4}\right)}^3$ and ${\left(\frac{2}{3}\right)}^3$.

Given: ${\left(\frac{-3}{4}\right)}^3\times \left[\frac{16}{27}-{\left(\frac{2}{3}\right)}^3\right]$.

We know that

${\left(-1\right)}^{\mathrm{n}}=\left\{\begin{array}{l}\mathrm{positive}\mathrm{number};\mathrm{if}\mathrm{n}\mathrm{is}\mathrm{an}\mathrm{even}\mathrm{number}\\ \mathrm{negative}\mathrm{number};\mathrm{if}\mathrm{n}\mathrm{is}\mathrm{an}\mathrm{odd}\mathrm{number}\end{array}\right.$

$$\begin{gathered} {\left(\frac{-3}{4}\right)}^3=\left(\frac{-3}{4}\right)\times \left(\frac{-3}{4}\right)\times \left(\frac{-3}{4}\right) \\ =\frac{-27}{64} \end{gathered}$$

$$\begin{gathered} {\left(\frac{2}{3}\right)}^3=\left(\frac{2}{3}\right)\times \left(\frac{2}{3}\right)\times \left(\frac{2}{3}\right) \\ =\frac{2\times 2\times 2}{3\times 3\times 3} \\ =\frac{8}{27} \end{gathered}$$

Step 2: Find the value of $\frac{16}{27}-{\left(\frac{2}{3}\right)}^3$.

$$\begin{gathered} \frac{16}{27}-{\left(\frac{2}{3}\right)}^3=\frac{16}{27}-\frac{8}{27} \\ =\frac{16-8}{27} \\ =\frac{8}{27} \end{gathered}$$

Step 3: Find the value of ${\left(\frac{-3}{4}\right)}^3\times \left[\frac{16}{27}-{\left(\frac{2}{3}\right)}^3\right]$.

$$\begin{gathered} {\left(\frac{-3}{4}\right)}^3\times \left[\frac{16}{27}-{\left(\frac{2}{3}\right)}^3\right]=\frac{-27}{64}\times \frac{8}{27} \\ =\frac{-\cancel{27}}{{\cancel{64}}^8}\times \frac{\cancel{8}}{\cancel{27}} \\ =\frac{-1}{8}\times \frac{1}{1} \\ =-\frac{1}{8} \end{gathered}$$

Hence, ${\left(\frac{-3}{4}\right)}^3\times \left[\frac{16}{27}-{\left(\frac{2}{3}\right)}^3\right]=-\frac{1}{8}$.