You know the square root sign? That’s really called a radical sign, and we need to call it a radical sign instead of a square root sign because we can use it to find much more than just square roots. With that having been said, let’s start off by looking at multiplying two radicals with an example.

$\sqrt{9} * \sqrt{16}$

Let’s simplify this and see if we can work out some sort of pattern.

$\sqrt{9} * \sqrt{16}$
$3 * 4 = 12$

I wonder if we can combine the radicals in some way and get the same answer?

$\sqrt{9} * \sqrt{16}$
$\sqrt{9 * 16}$
$\sqrt{144} = 12$

Yes, we can! Testing other examples will show that this pattern works for any two numbers, so we have found a pattern that we can use in more general terms:

$\sqrt{a} * \sqrt{b} = \sqrt{a*b}$

Sometimes we’re going to have more difficult radicals that need to be simplified, and we’re going to have to figure out a general formula for this. Let’s start with a simple example using a cube root:

$\sqrt[3]{x^{12}}$

To simplify this radical, we’re going to have to figure out what times itself times itself gives us x to the 12th power. Let’s break up what’s under the radical sign and see what we can figure out:

$\sqrt[3]{x^{12}}$
$\sqrt[3]{x^3 * x^3 * x^3 * x^3}$
$\sqrt[3]{x^3} * \sqrt[3]{x^3} * \sqrt[3]{x^3} * \sqrt[3]{x^3}$
$x * x * x * x$
$x^4$

This example seems to simplify some relationship between 3, 4, and 12. We know that 3*4 = 12, implying this definition/formula that you’re going to need to know, and that explains the situation above:

$\sqrt[b]{x^a} = x^{\frac{a}{b}}$

Testing other examples will help you to see that this formula is true. Let’s use this formula to solve a similar example:

$\sqrt[4]{81x^{20}}$
$\sqrt[4]{81} * \sqrt[4]{x^{20}}$
$3 * x^{\frac{20}{4}}$
$3x^5$

Dealing with radicals well takes a lot of practice, and you should spend a lot of time working on radicals now in Algebra 1 before moving on to more difficult math courses so that you’ll be ready to learn new material instead of having to come back to learn this well.