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Multiplication and Division Properties of Radicals

The Multiplication and Division Properties of Radicals can help you simplify expressions!

We can have radicals of any integer root, known as the $n$th root. Most problems will have $n=2$ or $n=3$. In other words, you will mostly be dealing with square roots $(\sqrt{\phantom{9}} \text{ or } \sqrt[2]{\phantom{9}})$ and cube roots $(\sqrt[3]{\phantom{9}})$.

Let's do an example to practice!

We use both the Multiplication and Division Properties of Radicals to solve this problem. First, we use the Division Property of Radicals since we are finding the square root of a fraction.

$$\begin{align} \sqrt{\dfrac{27}{48}}&=\dfrac{\sqrt{27}}{\sqrt{48}}, \text{ by the Division Property of Radicals}\\ \end{align}$$

We can further break down the $\sqrt{27}$ and $\sqrt{48}$ using the Multiplication Property of Radicals.

$$\begin{align} \dfrac{\sqrt{27}}{\sqrt{48}}&=\dfrac{\sqrt{9\cdot 3}}{\sqrt{16 \cdot 3}}\\ &=\dfrac{\sqrt{9}\cdot \sqrt{3}}{\sqrt{16}\cdot \sqrt{3}}, \text{ by the Multiplication Property of Radicals}\\ &=\dfrac{3\cdot \sqrt{3}}{4\cdot \sqrt{3}}\\ &=\dfrac{3\cdot \cancel{\sqrt{3}}}{4\cdot \cancel{\sqrt{3}}}\\ &=\dfrac{3}{4} \end{align}$$

So, our final answer is $\dfrac{3}{4}.$

Combine the Terms Under One Radical

Unlike adding and subtracting radicals, we can multiply two radicals and change the number underneath. Here's the rule: $$\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}$$

Multiplying two separate radicals with numbers in them is the same as multiplying those numbers under one radical. For example: $$\begin{align} \sqrt{16} \times \sqrt{9} &= \sqrt{16 \times 9}\\ 4 \times 3 &= \sqrt{144}\\ 12 &= 12 \end{align}$$ Although this example used perfect squares, no matter what expression is under the radical, either way we solve the problem, the answer comes out the same.

Does this property seem familiar? Maybe. This is kind of like the product rule for exponents.

The Steps

Sometimes it's easier to keep difficult numbers in radical form. In this case, you can:

1. Multiply the numbers from within the radicals together
2. Place the product under one radical symbol.
3. If the radicals were already multiplied by something, multiply those together and place them in front of the radical.

Step 1: Multiply the numbers from within the radicals together

Let's work with the expression $2\sqrt{7} \times 3\sqrt{13}$. We're going to multiply each part of each term together separately, then combine them again. Let's start with the numbers inside the radicals.

$$7 \times 13 = 91$$

Awesome!

Step 2: Place the product under one radical symbol

We already know that our product is $91$, so we need to place that under the radical. $$91\text{ becomes }\sqrt{91}$$

Step 3: Multiply the multipliers

Both radicals in our problem, $2\sqrt{7} \times 3\sqrt{13}$, have a coefficient. We multiplied the radicals, so now we need to multiply the coefficients.

$2 \times 3 = 6$. Great! Now we can place $6$ in front of the radical so we get: $$2\sqrt{7} \times 3\sqrt{13} = 6\sqrt{91}$$

Combine terms under one radical

Radical division follows the same rule as radical multiplication. Dividing two radicals means we can instead divide the two numbers under one radical. For $a\ge 0$ and $b>0$: $$\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$$

So, for example, we could evaluate the expression $\frac{\sqrt{64}}{\sqrt{16}}$ in two ways.

$$\begin{align}\frac{\sqrt{64}}{\sqrt{16}} &= \sqrt{\frac{64}{16}}\\ \frac{8}{4} &= \sqrt{4}\\ 2 &= 2 \end{align}$$

The left side of each equation shows that you can simplify the radicals before dividing. The right side shows you can divide before you simplify the radicals.

It's always important to be mindful of possible factors in the radicand that can help simplify expressions quicker.

The steps

It's good to keep messy numbers within their radicals for as long as possible. To do that while dividing, follow these steps:

1. Divide the numbers from within the radicals.
2. Place the result (fraction or not) under one radical symbol.
3. If the radicals were already multiplied by something (in other words, if they had a multiplier as part of their term), divide those numbers and place them in front of the radical. If only one radical has a multiplier, pretend the other is multiplied by $1$.

Let's try it with the expression, $$\frac{\sqrt{22}}{3\sqrt{11}}$$.

Step 1: Divide the numbers from within the radicals

The numbers in this expression won't have nice square roots at all. We're going to try to keep everything within the radicals for as long as possible.

$$\frac{22}{11} = 2$$

Good!

Step 2: Place the result under a radical symbol

So $2$ becomes $\sqrt{2}$.

Step 3: Divide the multipliers

In the original expression, $\sqrt{22}$ doesn't have a multiplier, but $\sqrt{11}$ is multiplied by $3$. If something doesn't have a multiplier, we can just pretend it's being multiplied by $1$, since that doesn't change anything.

Since our numerator is $1$ and our denominator is $3$, we get $\frac{1}{3}$ as our new multiplier. So our new expression is:

$$\begin{gather} \frac{1}{3} \times \sqrt{2}\\ \text{or}\\ \frac{\sqrt{2}}{3} \end{gather}$$

Here is a more direct way to think about it: $$\begin{align} \frac{\sqrt{22}}{3\sqrt{11}} &=\frac{1}{3} \cdot \frac{\sqrt{22}}{\sqrt{11}}\\ &=\frac{1}{3} \cdot \sqrt{\frac{22}{11}}\\ &=\frac{1}{3} \cdot \sqrt{2}\\ &=\frac{\sqrt{2}}{3} \end{align}$$

Sometimes, we may need to tackle a more complicated expression. There may be a funky radical expression in the numerator or denominator that may require more fancy footwork. Don't panic! You've got the skills to divide radical expressions, no matter how complicated they get.

If you've got a more complex-looking rational number in the denominator, you may need to rationalize the denominator by multiplying the fraction by the conjugate of the denominator. You'll learn more about this concept later, but something to keep in mind.

(Video) Multiplying and Dividing Radical Terms

This video by mathgirl915 shows you how to multiply and divide radical terms.

$\large{\text{Summary}}$

When multiplying radicals, first make sure that the degree of the radical is the same. Then, unlike addition and subtraction, the terms underneath the radical do not need to be the same! You can multiply the outer numbers, then put the product of the inner radical terms underneath the radical.

Suppose we want to multiply two radicals: $\sqrt3\times\sqrt{12}$. All we have to do is multiply the two numbers and put them under one radical. $$\begin{align} \sqrt{3}\times\sqrt{12}&=\sqrt{3\times12}\\ &=\sqrt{36}\\ &=6 \end{align}$$

Let's work through one that has numbers both inside and outside a radical.

$$8\sqrt{8} \times 6\sqrt{6}$$

We're going to multiply each part separately.

$$\begin{align} 8 \times 8 &= 48\\ \sqrt{8} \times \sqrt{6}&= \sqrt{48} \end{align}$$

We need to simplify $\sqrt{48}$

$$\sqrt{48}\\ \sqrt{16 \times 3}\\ 4\sqrt{3}$$

Now we combine these two sections. Anything outside of the radical needs to be multiplied.

$$48 \times 4\sqrt{3}\\ 192 \sqrt{3}$$