How do you simplify root3(150)*root3(20)?

Oct 21, 2016

$10 \sqrt[3]{3}$

Explanation:

Dealing with roots is the same as dealing with exponents. Except the exponents for roots are fractions. Like this:
$\sqrt[3]{x} = {x}^{\frac{1}{3}}$

That means we can apply the distributive property of exponents to roots as well:
$\sqrt[3]{x \cdot y} = {\left(x \cdot y\right)}^{\frac{1}{3}} = {x}^{\frac{1}{3}} \cdot {y}^{\frac{1}{3}}$

Now to our problem. If we factor both numbers we get that:
$150 = 5 \cdot 5 \cdot 3 \cdot 2$
$20 = 5 \cdot 2 \cdot 2$

so our problem becomes:
$\sqrt[3]{5 \cdot 5 \cdot 3 \cdot 2} \cdot \sqrt[3]{5 \cdot 2 \cdot 2}$

Then we can combine both under one root, like so:
$\sqrt[3]{5 \cdot 5 \cdot 3 \cdot 2 \cdot 5 \cdot 2 \cdot 2}$

As you can see there are three 5s and three 2s in there, meaning we can take them to the outside of the root like this:
$5 \cdot 2 \sqrt[3]{3}$

or even better:
$10 \sqrt[3]{3}$