Multiplying and Dividing Radical Expressions

As long as the indices are the same, we can multiply the radicands together using the following property.

Radical Expression Playlist on YouTube

Since multiplication is commutative, you can multiply the coefficients and the radicands together and then simplify.

Multiply.

Take care to be sure that the indices are the same before multiplying.  We will assume that all variables are positive.

Simplify.

Divide radicals using the following property.

Divide. (Assume all variables are positive.)

Rationalizing the Denominator

A simplified radical expression cannot have a radical in the denominator.  When the denominator has a radical in it, we must multiply the entire expression by some form of 1 to eliminate it.  The basic steps follow.

Rationalize the denominator: 

Multiply numerator and denominator by the 5th root of of factors that will result in 5th powers of each factor in the radicand of the denominator.

Notice that all the factors in the radicand of the denominator have powers that match the index.  This was the desired result, now simplify.

Rationalize the denominator.

This technique does not work when dividing by a binomial containing a radical.  A new technique is introduced to deal with this situation.

Rationalize the denominator: 

Multiply numerator and denominator by the conjugate of the denominator.

And then simplify. The goal is to eliminate all radicals from the denominator.

Rationalize the denominator.

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Simplifying Radical Expressions

Now when working with square roots and variables we should be a bit careful.

Playlist: Steps for Simplifying Radical Expressions

The variable could represent a positive or negative number so we must ensure that it is positive by making use of the absolute value.

To avoid this technicality many textbooks state, at this point, that we assume all variables are positive. If not, use the absolute values as in the following problems.

Simplify. (Assume variables could be negative.)

To avoid many of the technicalities when working with nth roots we will assume, from this point on, that all the variables are positive.

Simplify. (Assume all variables represent positive numbers.)

Students often try to simplify the previous problem. Be sure to understand what makes the following problems all different.

       

The property:

says that we can simplify when the operation is multiplication.  There is no corresponding property for addition or subtraction.

Video Examples on YouTube: