Question

Simplify:

(i) $({16}^{-1/5}{)}^{5/2}$

(ii) $\sqrt[5]{{\left(32\right)}^{-3}}$

(iii) $\sqrt[3]{(343{)}^{-2}}$

(iv) $(0.001{)}^{1/3}$

(v) $\frac{(25{)}^{3/2}\times (243{)}^{3/5}}{(16{)}^{5/4}\times (8{)}^{4/3}}$

(vi) ${\left(\frac{\sqrt{2}}{5}\right)}^8÷{\left(\frac{\sqrt{2}}{5}\right)}^{13}$

(vii) ${\left(\frac{5^{-1}\times 7^2}{5^2\times 7^{-4}}\right)}^{7/2}\times {\left(\frac{5^{-2}\times 7^3}{5^3\times 7^{-5}}\right)}^{-5/2}$

Answer 1

(1) Given

By using law of rational exponents we have

Hence the value of is
(ii) $\sqrt[5]{{\left(32\right)}^{-3}}$

$$\begin{gathered} =\sqrt[5]{{\left(\frac{1}{32}\right)}^3} \\ ={\left(\frac{1}{32}\right)}^{\frac{3}{5}} \\ ={\left(\frac{1}{{\left(2\right)}^5}\right)}^{\frac{3}{5}} \\ ={\left(\frac{1}{2}\right)}^{5\times \frac{3}{5}} \end{gathered}$$
$$\begin{gathered} ={\left(\frac{1}{2}\right)}^3 \\ =\left(\frac{1}{8}\right) \end{gathered}$$

(iii) Given

Hence the value of is

(iv) Given

The value of is

(v) Given

Hence the value of is

(vi) Given. So,

By using the law of rational exponents

Hence the value of is

(vii) Given . So,

Hence the value of is