# How do you simplify sqrt(16/27)*sqrt(5/3)?

Feb 5, 2018

color(purple)((4/9) * sqrt5

#### Explanation:

To simplify $\sqrt{\frac{16}{27}} \cdot \sqrt{\frac{5}{3}}$

According to theory of indices,

${a}^{m} \cdot {b}^{m} = {\left(a \cdot b\right)}^{m}$. Also, ${a}^{m} \cdot {a}^{n} = {a}^{m + n}$

Using the above properties,

sqrt(16/27) * sqrt5/3) = (16/27)^(1/2) * (5/3)^(1/2)

$\implies {\left(\left(\frac{16}{27}\right) \left(\frac{5}{3}\right)\right)}^{2} = {\left(\frac{16 \cdot 5}{27 \cdot 3}\right)}^{\frac{1}{2}}$

=> ((4 * 4 * 5) / (9 * 9))^(1/2) = color(purple)((4/9) * sqrt5

Feb 5, 2018

The answer is $\frac{4 \sqrt{5}}{9}$, or about $0.994$.

#### Explanation:

Radicals that are multiplied together can be condensed together if you multiply their radicands (the stuff under the radical):

$\sqrt{\frac{16}{27}} \cdot \sqrt{\frac{5}{3}}$

$\sqrt{\frac{16 \cdot 5}{27 \cdot 3}}$

$\sqrt{\frac{80}{81}}$

$\frac{\sqrt{80}}{\sqrt{81}}$

$\frac{\sqrt{16 \cdot 5}}{9}$

$\frac{4 \sqrt{5}}{9} \approx 0.99381 \ldots$