Question

Simplify and express the answer in exponential form :
$\frac{9^3\times a^7\times {15}^2\times 80\times b^5}{3^4\times a^5\times 5^3\times b^2}$

Answer 1

$\frac{9^3\times a^7\times {15}^2\times 80\times b^5}{3^4\times a^5\times 5^3\times b^2}$

$=\frac{(3^2{)}^3\times a^7\times (3\times 5{)}^2\times (2\times 5\times 2^3)\times b^5}{3^4\times a^5\times 5^3\times b^2}$

$\frac{3^6\times a^7\times 3^2\times 5^2\times 2^4\times 5^1\times b^5}{3^4\times a^5\times 5^3\times b^2}$ since $(a^m{)}^n=a^{mn}$ and $(ab{)}^m=a^m{b}^m$

$=\frac{3^{6+2}\times a^7\times 5^{2+1}\times 2^4\times b^5}{3^4\times a^5\times 5^3\times b^2}$ Since $a^m{a}^n=a^{m+n}$

$=\frac{3^8\times a^7\times 5^3\times 2^4\times b^5}{3^4\times a^5\times 5^3\times b^2}$

$=3^{8-4}\times a^{7-5}\times 5^{3-3}\times 2^4\times b^{5-2}$ Since $\frac{a^m}{a^n}=a^{m-n}$

$=3^4\times a^2\times 5^0\times 2^4\times b^3$

$=3^4{a}^2\times 2^4{b}^3$ since $5^0=1$

$=(3\times 2{)}^4{a}^2{b}^3$ since $a^m{b}^m=(ab{)}^m$

$=6^4{a}^2{b}^3=1296a^2{b}^3$