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)

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

$=3 \times a^{40} \times b^{30} \times 5 \times a^{6} \times b^{6}$

$=15 \times a^{40} \times a^{6} \times b^{30} \times b^{6}$

$=15 \times a^{40+6} \times b^{30+6} \quad\left[a^{m} \times a^{n}=a^{m+n}\right]$

$=15 a^{46} b^{36}$

(ii)

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

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

$=8 \times x^{-6} \times y^{9}$

$=8 x^{-6} y^{9}$

(iii)

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

$=\frac{4 \times 10^{7} \times 6 \times 10^{-5}}{8 \times 10^{4}}$

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

$=\frac{24 \times 10^{2}}{8 \times 10^{4}}$

$=\frac{24 \times 10^{2} \times 10^{-4}}{8}$

$=3 \times 10^{2+(-4)}$

$=3 \times 10^{-2}$

$=\frac{3}{100}$

(iv)

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

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

$=\frac{-20 \times a^{1} \times a^{1} \times b^{2} \times b^{3}}{10 a^{2} b^{2}}$

$=\frac{-20 \times a^{1+1} \times b^{2+3}}{10 a^{2} b^{2}}$

$=-2 \times a^{2} \times b^{5} \times a^{-2} \times b^{-2}$

$=-2 \times a^{2+(-2)} \times b^{5+(-2)}$

$=-2 \times a^{0} \times b^{3}$

$=-2 b^{3}$

(v)

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

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

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

(vi)

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

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

$=\frac{a^{(18 n-54)}}{a^{2 n-4}}$

$=a^{(18 n-54)} \times a^{-(2 n-4)}$

$=a^{18 n-54} \times a^{-2 n+4}$

$=a^{18 n-54-2 n+4}$

$=a^{16 n-50}$