Question

Simplify:
(i) $(-4{)}^5÷(-4{)}^8$

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

(iii) $(-3{)}^4\times {\left(\frac{5}{3}\right)}^4$

(iv) ${\left(\frac{2}{3}\right)}^5\times {\left(\frac{3}{4}\right)}^2\times {\left(\frac{1}{5}\right)}^2$

(v) $(3^{-7}÷3^{10})\times 3^{-5}$

(vi) $\frac{2^6\times 3^2\times 2^3\times 3^7}{2^8\times 3^6}$

(vii) $y^{a-b}\times y^{b-c}\times y^{c-a}$

Answer 1

(i) $(-4{)}^5÷(-4{)}^8$

$\Rightarrow (-4{)}^5\times \frac{1}{(-4{)}^8}$

$\Rightarrow (-1)\times (4{)}^5\times \frac{1}{(4{)}^8}$

$\Rightarrow (-1)\times \frac{1}{(4{)}^3}$

$\Rightarrow -\frac{1}{64}$

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

$\Rightarrow {\left(\frac{1}{8}\right)}^2$

$\Rightarrow \frac{1}{64}$

(iii) $(-3{)}^4\times {\left(\frac{5}{3}\right)}^4$

$\Rightarrow (3{)}^4\times \left(\frac{5^4}{3^4}\right)$

$\Rightarrow 5^4$

$\Rightarrow 625$

(iv) ${\left(\frac{2}{3}\right)}^5\times {\left(\frac{3}{4}\right)}^2\times {\left(\frac{1}{5}\right)}^2$

$\Rightarrow \left(\frac{2^5}{3^5}\right)\times \left(\frac{3^2}{4^2}\right)\times \left(\frac{1}{5^2}\right)$

$\Rightarrow \left(\frac{32}{3^3\times 3^2}\right)\times \left(\frac{3^2}{16}\right)\times \left(\frac{1}{25}\right)$

$\Rightarrow \left(\frac{2}{3^3}\right)\times \left(\frac{1}{25}\right)$

$\Rightarrow \left(\frac{2}{27}\right)\times \left(\frac{1}{25}\right)$

$\Rightarrow \frac{2}{675}$

(v) $(3^{-7}÷3^{10})\times 3^{-5}$

$\Rightarrow (\frac{1}{3^7}÷3^{10})\times \frac{1}{3^5}$

$\Rightarrow (\frac{1}{3^7}\times \frac{1}{3^{10}})\times \frac{1}{3^5}$

$\Rightarrow (\frac{1}{3^{17}}\times \frac{1}{3^5})$

$\Rightarrow \frac{1}{3^{22}}$

(vi) $\frac{2^6\times 3^2\times 2^3\times 3^7}{2^8\times 3^6}$

$\Rightarrow \frac{2^6\times 3^2\times 2^2\times 2\times 3^6\times 3}{2^6\times 2^2\times 3^6}$

$\Rightarrow \frac{1\times 3^2\times 2^2\times 2\times 3}{1\times 1}$

$\Rightarrow 216$

(vii) $y^{a-b}\times y^{b-c}\times y^{c-a}$

$\Rightarrow \frac{y^a}{y^b}\times \frac{y^b}{y^c}\times \frac{y^c}{y^a}$

$\Rightarrow 1$

Hence, this is the answer.