Simplifying a ratio with powers

Statement

Simplify, factoring and using the properties of powers: $\dfrac{3072}{729}$

Solution

Factor numerator and denominator: $3072=2^{10}\cdot 3$ (indeed $2^{10}=1024$ and $1024\cdot 3=3072$) and $729=3^{6}$. Hence $\frac{3072}{729}=\frac{2^{10}\cdot 3}{3^{6}}=\frac{2^{10}}{3^{5}}=\frac{1024}{243}\approx 4.21.$

$\boxed{\dfrac{3072}{729}=\dfrac{2^{10}}{3^{5}}=\dfrac{1024}{243}}$