Question

Simplify

(a)${\left(\frac{1}{2}{y}^2\right)}^3{\left(\frac{2}{3}y\right)}^2$

(b)${\left(\frac{1}{2}{j}^2\right)}^5{\left(\frac{2}{5}{j}^3\right)}^2$

Short answer

Simplified expressions are

(a)$\frac{y^8}{18}$

(b)$\frac{j^{16}}{200}$

Step-by-step solution

part (a) Step 1 Given information 

${\left(\frac{1}{2}{y}^2\right)}^3{\left(\frac{2}{3}y\right)}^2$

part (a) Step 2

$$\begin{gathered} {\left(\frac{1}{2}{y}^2\right)}^3{\left(\frac{2}{3}y\right)}^2={\left(\frac{1}{2}\right)}^3{\left(y^2\right)}^3{\left(\frac{2}{3}\right)}^2{\left(y\right)}^2 \\ =\frac{1}{2^3}\times y^{2\times 3}\times \frac{2^2}{3^2}\times y^2 \\ =\frac{1}{8}\times y^6\times \frac{4}{9}\times y^2 \\ =\frac{1}{18}\times y^{6+2} \\ =\frac{y^8}{18} \end{gathered}$$

part (b) Step 1 Given information 

${\left(\frac{1}{2}{j}^2\right)}^5{\left(\frac{2}{5}{j}^3\right)}^2$

part (b) Step 2

$$\begin{gathered} {\left(\frac{1}{2}{j}^2\right)}^5{\left(\frac{2}{5}{j}^3\right)}^2={\left(\frac{1}{2}\right)}^5{\left(j^2\right)}^5{\left(\frac{2}{5}\right)}^2{\left(j^3\right)}^2 \\ =\frac{1}{2^5}\times j^{2\times 5}\times \frac{2^2}{5^2}\times j^{3\times 2} \\ =\frac{1}{32}\times j^{10}\times \frac{4}{25}\times j^6 \\ =\frac{1}{200}\times j^{10+6} \\ =\frac{j^{16}}{200} \end{gathered}$$