d: =5^10*5^4/5^12=5^2
e: \(=\dfrac{2^{19}\cdot3^9+3^9\cdot2^{18}\cdot5}{2^{19}\cdot3^9+3^{10}\cdot2^{20}}\)
\(=\dfrac{2^{18}\cdot3^9\left(2+5\right)}{2^{19}\cdot3^9\left(1+3\cdot2\right)}=\dfrac{1}{2}\)
`@` `\text {Ans}`
`\downarrow`
`d)`
\(\dfrac{5^{10}\cdot25^2}{125^4}\)
`=`\(\dfrac{5^{10}\cdot\left(5^2\right)^2}{\left(5^3\right)^4}=\dfrac{5^{10}\cdot5^4}{5^{12}}=\dfrac{5^{14}}{5^{12}}=5^2\)
`e)`
\(\dfrac{2^{19}\cdot27^3+15\cdot4^9\cdot9^4}{6^9\cdot2^{10}+12^{10}}\)
`=`\(\dfrac{2^{19}\cdot3^9+3\cdot5\cdot2^{18}\cdot3^8}{2^9\cdot3^9\cdot2^{10}+2^{20}\cdot3^{10}}\)
`=`\(\dfrac{2^{19}\cdot3^9+5\cdot2^{18}\cdot3^9}{2^{19}\cdot3^9+2^{20}\cdot3^{10}}\)
`=`\(\dfrac{2^{18}\cdot3^9\left(2+5\right)}{2^{19}\cdot3^9\left(1+2\cdot3\right)}\)
`=`\(\dfrac{2^{18}\cdot3^9}{2^{19}\cdot3^9}=\dfrac{1}{2}\)
