Multiplying and Dividing Radicals

6.2 - Multiply and Divide Radicals

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A reminder of the properties of roots

It's important to remember that the product property works in both directions.

Now that we know about roots with an index other than 2

We can only apply the product property to roots with the same index.

Multiple Choice

Which of these expressions cannot be simplified using the product property?

• $$\sqrt{3}\cdot\sqrt{40}$$

• $$^3\sqrt{8}\cdot\ \sqrt{64}$$

• $$\sqrt{6}\cdot\sqrt{72}$$

• $$\sqrt{5}\cdot\sqrt{5}$$

Let's practice

Multiple Choice

Simplify the expression

$$\sqrt{27}\cdot\sqrt{5}$$  

• $$15\sqrt{3}$$  

• $$3\sqrt{35}$$  

• $$3\sqrt{15}$$  

• $$3\sqrt{5}$$  

Multiple Choice

Simplify the expression

$$\sqrt{8}\cdot\sqrt{12}$$  

• $$4\sqrt{3}$$  

• $$6\sqrt{16}$$  

• Cannot be simplified

• $$4\sqrt{6}$$  

Multiple Select

Which expressions are equivalent to

$$6\sqrt{3}$$  ? Check all that apply.

• $$\sqrt{9}\cdot\sqrt{12}$$  

• $$\sqrt{18}\cdot\sqrt{6}$$  

• $$\sqrt{18}$$  

• $$\sqrt{6}\cdot\sqrt{9}$$  

What if there is an expression outside the radical? Can we still multiply? Yes!

• Multiply the factors outside the radical. The product stays outside the radical

• Multiply the factors inside the radical. Simplify if possible. If necessary, simplify outside the radical as well.

Example:

You try

Multiple Choice

Simplify the expression

 

$$2\sqrt[3]{9}\cdot5\sqrt[3]{-24}$$

•   $$-240$$

•   $$-60$$

•   $$10\sqrt[3]{-6}$$

•   $$-6\sqrt[3]{10}$$

Binomial Example

Math Response

Simplify the expression $$\left(4-\sqrt[]{5}\right)^2$$

Type answer here

Correct Answer: 21-8\sqrt{5}

Deg^°

Rad

For a radical expression to be considered simplified, three things must be true:

• No perfect square factors (other than 1) under the radical

• No fractions under the radical

• No radicals in the denominator of a fraction

Example 1

Example 2

But if we cant simplify the radical using any of these methods we will rationalize the denominator

We multiply the numerator and denominator by the same number, and we pick that number on purpose so that the denominator simplifies to an integer

Example 3

You try

Multiple Choice

Simplify the expression

$$\frac{3}{2\sqrt[]{5}}$$  

• $$\frac{15}{\sqrt[]{2}}$$  

• $$\frac{\sqrt[]{15}}{2}$$  

• $$\frac{3\sqrt[]{5}}{10}$$  

• $$\frac{\sqrt[]{3}}{2}$$  

Poll

Select the image that most closely represents how well you feel you can multiply and divide radicals: