Question

Simplify the following:
a) ${\left(a^8\times a^5\right)}^0$
b) ${\left(b^2\right)}^4\times b^0$
c) $-\frac{a^{-3}{b}^{10}{c}^8\times b{c}^8}{{\left(b^{-10}\times c^6\right)}^4}$
d) ${\left[\frac{-a^{-5}}{-a^{10}{b}^{-9}{c}^{-7}\times \left(a^{-15}\times c^0\right)}\right]}^2$

Answer 1

$a){\left(a^8\times a^{-5}\right)}^0=1\left[\because a^0=1\right]$

$b){\left(b^2\right)}^4\times b^0=b^8\times b^0\left[{\left(a^n\right)}^m=a^{mn}\right]$

$=b^8\times 1\left[a^0=1\right]$

$c)\frac{-a^{-3}{b}^{10}{c}^8\times b{c}^8}{{\left(b^{-10}\times c^6\right)}^4}$

$=\frac{-a^{-3}\cdot b^{10}\cdot c^8\times b^1\times c^8}{b^{-40}\times c^{24}}\left[\because {\left(a^m\right)}^n=a^{mn}\right]$

$=\frac{-a^{-3}\times b^{11}\times c^{16}}{b^{-40}\times c^{24}}\left[\because \frac{a^m}{a^n}=a^{m-n}\right]$

$=-a^{-3}\times b^{11+40}\times c^{16-24}$

$=-a^{-3}\times b^{51}\times c^{-8}$

$=\frac{-b^{51}}{a^3{c}^8}$

$d){\left[\frac{-a^{-5}}{-a^{10}{b}^{-9}{c}^{-7}\times \left(a^{-15}\times c^0\right)}\right]}^2$

$=\frac{-a^{-10}}{-a^{20}\times b^{-18}\times c^{-14}\times a^{-30}\times c^0}$

$=\frac{-a^{-10}}{-a^{20-30}\times b^{-18}\times c^{-14+0}}$

$=\frac{a^{-10}}{a^{-10}\times b^{-18}\times c^{-14}}$

$=b^{18}{c}^{14}$