# Simplify the following:

Question:

Simplify the following:

(i) $3\left(a^{4} b^{3}\right)^{10} \times 5\left(a^{2} b^{2}\right)^{3}$

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

(iii) $\frac{\left(4 \times 10^{7}\right)\left(6 \times 10^{-5}\right)}{8 \times 10^{4}}$

(iv) $\frac{4 a b^{2}\left(-5 a b^{3}\right)}{10 a^{2} b^{2}}$

(v) $\left(\frac{x^{2} y^{2}}{a^{2} b^{3}}\right)^{n}$

(vi) $\frac{\left(a^{3 n}-9\right)^{6}}{a^{2 n-4}}$

Solution:

(i) $3\left(a^{4} b^{3}\right)^{10} \times 5\left(a^{2} b^{2}\right)^{3}$

$=3\left(a^{40} b^{30}\right) \times 5\left(a^{6} b^{6}\right)$

$=15\left(a^{46} b^{36}\right)$

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

$\left(2^{3} \times^{-2 \times 3} y^{3 \times 3}\right)=8 x^{-6} y^{9}$

(iii) $\frac{\left(4 \times 10^{7}\right)\left(6 \times 10^{-5}\right)}{8 \times 10^{4}}$

$\frac{\left(4 \times 10^{7}\right)\left(6 \times 10^{-5}\right)}{8 \times 10^{4}}$

$=\frac{\left(24 \times 10^{7} \times 10^{-5}\right)}{8 \times 10^{4}}$

$=\frac{\left(24 \times 10^{7-5}\right)}{8 \times 10^{4}}$

$=\frac{\left(24 \times 10^{2}\right)}{8 \times 10^{4}}$

$=\frac{3}{100}$

(iv) $\frac{4 a b^{2}\left(-5 a b^{3}\right)}{10 a^{2} b^{2}}$

$\frac{4 a b^{2}\left(-5 a b^{3}\right)}{10 a^{2} b^{2}}$

$=-\frac{20 a^{2} b^{5}}{10 a^{2} b^{2}}=-2 b^{3}$

(v) $\left(\frac{x^{2} y^{2}}{a^{2} b^{3}}\right)^{n}$

$\left(\frac{x^{2} y^{2}}{a^{2} b^{3}}\right)^{n}$

$=\frac{x^{2 n} y^{2 n}}{a^{2 n} b^{3 n}}$

(vi) $\frac{\left(a^{3 n}-9\right)^{6}}{a^{2 n-4}}$

$\frac{\left(a^{3 n}-9\right)^{6}}{a^{2 n-4}}$

$=\frac{\mathrm{a}^{18 \mathrm{n}-54}}{\mathrm{a}^{2 \mathrm{n}-4}}=\mathrm{a}^{18 \mathrm{n}-2 \mathrm{n}-54+4}=\mathrm{a}^{16 \mathrm{n}-50}$