5. Multiplication and Division of Radicals
When multiplying expressions containing radicals, we use the law 
, along with normal procedures of algebraic multiplication.
Example 1
On this page...
(a) 
Answer:
(b) 
Answer:
Example 2
(a) 
Answer:
In this case, we needed to find the largest cube that divides into 24. (The cubes are the numbers 13 = 1, 23 = 8, 33 = 27, 43 = 64, ...)
(b) 
Answer:
Example 3
(a) 
Answer:
(b) 
Answer:
Division of Radicals (Rationalizing the Denominator)
This process is also called "rationalising the denominator" since we remove all irrational numbers in the denominator of the fraction.
This is important later when we come across Complex Numbers.
Reminder: From earlier algebra, you will recall the difference of squares formula:
(a + b)(a − b) = a2 − b2
We will use this formula to rationalize denominators.
Example
Simplify: 
Answer
The question requires us to divide 1 by (√3 − √2).
We need to multiply top and bottom of the fraction by the conjugate of (√3 − √2).
The conjugate is easily found by reversing the sign in the middle of the radical expression. In this case, our minus becomes plus. So the conjugate of (√3 − √2) is (√3 + √2).
After we multiply top and bottom by the conjugate, we see that the denominator becomes free of radicals (in this case, the denominator has value 1).
Historical Note
In the days before calculators, it was important to be able to rationalize denominators. Using logarithm tables, it was very troublesome to find the value of expressions like our example above.
Now that we use calculators, it is not so important to rationalize denominators.
However, rationalizing denominators is still used for several of our algebraic techniques (see especially Complex Numbers), and is still worth learning.
Exercises
Q1 
Q2 
Q3 
Q4 
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