# How do you simplify sqrt32*sqrt144?

$48 \sqrt{2.}$
We use prime factorisation of $32 = {2}^{5}$ and $144 = {2}^{4} \cdot {3}^{2.}$
Hence the expression, $= \sqrt{32} \cdot \sqrt{144} ,$
$= {\left({2}^{5}\right)}^{\frac{1}{2}} \cdot {\left({2}^{4} \cdot {3}^{2}\right)}^{\frac{1}{2}}$=${\left({2}^{4} \cdot 2\right)}^{\frac{1}{2}} \cdot {\left({2}^{4} \cdot {3}^{2}\right)}^{\frac{1}{2}}$
$= \left\{{\left({2}^{4}\right)}^{\frac{1}{2}} {2}^{\frac{1}{2}}\right\} \left\{{\left({2}^{4}\right)}^{\frac{1}{2}} \cdot {\left({3}^{2}\right)}^{\frac{1}{2}}\right\}$=${2}^{4 \cdot \frac{1}{2}} \cdot {2}^{\frac{1}{2}} \cdot {2}^{4 \cdot \frac{1}{2}} \cdot {3}^{2 \cdot \frac{1}{2}} = {2}^{2} \cdot {2}^{\frac{1}{2}} \cdot {2}^{2} \cdot {3}^{1} = 4 \cdot \sqrt{2} \cdot 4 \cdot 3 = 48 \sqrt{2.}$