11.4: Multiplication and Division of Radicals
What if you knew that the area of a rectangular mirror is
Guidance
To multiply radicands, the roots must be the same.
Example A
Simplify
Solution:
Dividing radicals is more complicated. A radical in the denominator of a fraction is not considered simplified by mathematicians. In order to simplify the fraction, you must rationalize the denominator.
To rationalize the denominator means to remove any radical signs from the denominator of the fraction using multiplication.
Remember:
Example B
Simplify
Solution:
We must clear the denominator of its radical using the property above. Remember, what you do to one piece of a fraction, you must do to all pieces of the fraction.
RealWorld Radicals
Example C
A pool is twice as long as it is wide and is surrounded by a walkway of uniform width of 1 foot. The combined area of the pool and the walkway is 400 squarefeet. Find the dimensions of the pool and the area of the pool.
Solution:
 Make a sketch.

Let
x= the width of the pool. 
Write an equation.
Area=length⋅width
Combined length of pool and walkway
Combined width of pool and walkway
Since the combined area of the pool and walkway is
4. Solve the equation:
Use the quadratic formula.
5. We can disregard the negative solution since it does not make sense in this context. Thus, we can check our answer of 12.65 by substituting the result into the area formula.
The answer checks out.
Guided Practice
Simplify
Solution:
In this case, we need to make the number inside the cube root a perfect cube. We need to multiply the numerator and the denominator by
Practice
Sample explanations for some of the practice exercises below are available by viewing the following videos. Note that there is not always a match between the number of the practice exercise in the videos and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both
CK12 Basic Algebra: How to Rationalize a Denominator (10:18)
Multiply the following expressions.

6√(10−−√+8√) 
(a√−b√)(a√+b√) 
(2x√+5)(2x√+5)
Rationalize the denominator.

715−−√ 
910−−√ 
2x5√x 
5√3√y  The volume of a spherical balloon is
950cm3 . Find the radius of the balloon. (Volume of a sphere=43πR3 )  A rectangular picture is 9 inches wide and 12 inches long. The picture has a frame of uniform width. If the combined area of picture and frame is
180in2 , what is the width of the frame?  The volume of a soda can is
355 cm3 . The height of the can is four times the radius of the base. Find the radius of the base of the cylinder.
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Here you'll learn how to multiply and divide by radicals, as well as how to rationalize denominators.