Multiplying then Simplifying Radical Expressions

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  • 0:02 Product Rule for Radicals
  • 0:55 Multiplying Radical…
  • 2:00 Simplifying Radical…
  • 3:33 Lesson Summary
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Lesson Transcript
Instructor: Jennifer Beddoe

Jennifer has an MS in Chemistry and a BS in Biological Sciences.

Multiplying 2 or more radical expressions uses the same principles as multiplying polynomials, with a few extra rules for dealing with the radicals. This lesson will teach you how to multiply and then simplify radical expressions.

Product Rule for Radicals

When multiplying expressions that contain radicals, it can be most helpful to remember the product rule for radicals. This rule states that if you have two terms that contain radicals, you can combine them under the same radical symbol if the index is the same. The index of a radical is the superscript number in the 'V' part of the radical.


This radical has an index of 3 and it is called the cube root. If there is no number there, the index is assumed to be 2. That is referred to as the square root. Cube roots and square roots are the most common radicals.


Here is an example of how the product rule for radicals works.


You can see that by combining the terms under the radicals, the multiplication problem looks much easier to solve.

Multiplying Radical Expressions

By using the product rule to combine terms under the same radical symbol, it's easy to take the next step and multiply those terms together. In our previous example, we can quickly see that we need to multiply 5 and 3, which is 15. So the final answer to the problem is the square root of 15.


When multiplying 2 or more radical expressions, as long as each term has the same index, you can combine the terms under each radical and then multiply using the multiplication rule for exponents, which is that when terms have the same base, just add the exponents in order to multiply.

Let's try an example.



Since the index of each term is 4, you can combine each term under one radical symbol because of the product rule.


Then you can multiply each term together to get the answer:


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