Question

Assuming that x, y, z are positive real numbers, simplify each of the following:

(i) ${\left(\sqrt{x^{-3}}\right)}^5$

(ii) $\sqrt{x^3{y}^{-2}}$

(iii) $(x^{-2/3}{y}^{-1/2}{)}^2$

(iv) $(\sqrt{x}{)}^{-2/3}\sqrt{y^4}÷\sqrt{x{y}^{-1/2}}$

(v) $\sqrt[5]{243x^{10}{y}^5{z}^{10}}$

(vi) ${\left(\frac{x^{-4}}{y^{-10}}\right)}^{5/4}$

(vii) ${\left(\frac{\sqrt{2}}{\sqrt{3}}\right)}^5{\left(\frac{6}{7}\right)}^2$

Answer 1

We have to simplify the following, assuming thatare positive real numbers

(i) Given

As x is positive real number then we have

Hence the simplified value of is

(ii) Given

As x and y are positive real numbers then we can write

By using law of rational exponents we have

Hence the simplified value of is

(iii) Given

As x and y are positive real numbers then we have

By using law of rational exponents we have

By using law of rational exponents we have

$$\begin{gathered} {\left(x^{\frac{-2}{3}}{y}^{{}^{\frac{-1}{2}}}\right)}^2=\frac{1}{x^{\frac{2}{3}+\frac{2}{3}}}\times \frac{1}{y^{\frac{1}{2}+\frac{1}{2}}} \\ =\frac{1}{x^{\frac{4}{3}}}\times \frac{1}{y^{\frac{2}{2}}}=\frac{1}{x^{\frac{4}{3}}}\times \frac{1}{y} \\ =\frac{1}{x^{\frac{4}{3}}y} \end{gathered}$$

Hence the simplified value of is .

$$\begin{gathered} \left(\mathrm{iv}\right){\left(\sqrt{x}\right)}^{-\frac{2}{3}}\sqrt{y^4}÷\sqrt{x{y}^{-\frac{1}{2}}} \\ ={\left(x^{\frac{1}{2}}\right)}^{-\frac{2}{3}}{\left(y^4\right)}^{\frac{1}{2}}÷{\left(x\times y^{-\frac{1}{2}}\right)}^{\frac{1}{2}} \\ =\frac{x^{{\displaystyle \frac{1}{2}}\times -{\displaystyle \frac{2}{3}}}\times y^{4\times {\displaystyle \frac{1}{2}}}}{x^{{\displaystyle \frac{1}{2}}}\times y^{-{\displaystyle \frac{1}{2}}\times {\displaystyle \frac{1}{2}}}} \\ =\frac{x^{-{\displaystyle \frac{1}{3}}}\times y^2}{x^{{\displaystyle \frac{1}{2}}}\times y^{-{\displaystyle \frac{1}{4}}}} \\ \mathrm{by}\mathrm{using}\mathrm{the}\mathrm{law}\mathrm{of}\mathrm{rational}\mathrm{exponents},a^m÷a^n=a^{m-n},\mathrm{we}\mathrm{have} \\ {x}^{-\frac{1}{3}-\frac{1}{2}}\times y^{2+\frac{1}{4}} \\ =x^{-\frac{5}{6}}\times y^{\frac{9}{4}} \\ =\frac{y^{{\displaystyle \frac{9}{4}}}}{x^{{\displaystyle \frac{5}{6}}}} \\ \left(\mathrm{v}\right).\sqrt[5]{243x^{10}{y}^5{z}^{10}} \\ ={\left(243\times x^{10}\times y^5\times z^{10}\right)}^{\frac{1}{5}} \\ ={\left(243\right)}^{\frac{1}{5}}\times {\left(x^{10}\right)}^{\frac{1}{5}}\times {\left(y^5\right)}^{\frac{1}{5}}\times {\left(z^{10}\right)}^{\frac{1}{5}} \\ ={\left(3^5\right)}^{\frac{1}{5}}\times x^{10\times \frac{1}{5}}\times y^{5\times \frac{1}{5}}\times z^{10\times \frac{1}{5}} \\ =3\times x^2\times y\times z^2 \\ =3x^{2}y{z}^2 \\ \left(\mathrm{vi}\right){\left(\frac{x^{-4}}{y^{-10}}\right)}^{\frac{5}{4}} \\ =\frac{{\left(x^{-4}\right)}^{{\displaystyle \frac{5}{4}}}}{{\left(y^{-10}\right)}^{{\displaystyle \frac{5}{4}}}} \\ =\frac{x^{-4\times {\displaystyle \frac{5}{4}}}}{y^{-10\times {\displaystyle \frac{5}{4}}}} \\ =\frac{x^{-5}}{y^{-{\displaystyle \frac{25}{2}}}} \\ =\frac{y^{{\displaystyle \frac{25}{2}}}}{x^5} \end{gathered}$$

(vii) ${\left(\frac{\sqrt{2}}{\sqrt{3}}\right)}^5{\left(\frac{6}{7}\right)}^2$

$$\begin{gathered} {\left(\frac{\sqrt{2}}{\sqrt{3}}\right)}^5{\left(\frac{6}{7}\right)}^2 \\ ={\left(\frac{\sqrt{2}}{\sqrt{3}}\right)}^{2+2+1}{\left(\frac{6}{7}\right)}^2 \\ ={\left(\frac{\sqrt{2}}{\sqrt{3}}\right)}^2\times {\left(\frac{\sqrt{2}}{\sqrt{3}}\right)}^2\times {\left(\frac{\sqrt{2}}{\sqrt{3}}\right)}^1\times {\left(\frac{6}{7}\right)}^2 \\ =\left(\frac{2}{3}\right)\times \left(\frac{2}{3}\right)\times {\left(\frac{\sqrt{2}}{\sqrt{3}}\right)}^1\times {\left(\frac{6}{7}\right)}^2 \end{gathered}$$
$$\begin{gathered} =\left(\frac{2}{3}\right)\times \left(\frac{2}{3}\right)\times {\left(\frac{\sqrt{2}}{\sqrt{3}}\right)}^1\times {\left(\frac{6}{7}\right)}^2 \\ =\frac{16\sqrt{2}}{49\sqrt{3}} \\ =\sqrt{\frac{512}{7203}}={\left(\frac{512}{7203}\right)}^{\frac{1}{2}} \end{gathered}$$