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5. Multiplication and Division of Radicals

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Rationalizing the Denominator

When multiplying expressions containing radicals, we use the following law, along with normal procedures of algebraic multiplication.


Example 1

(a) `sqrt5sqrt2`

(b) `sqrt33sqrt3`

Continued below

Example 2

(a) `root(3)6root(3)4`

(b) `root(5)(8a^3b^4)root(5)(8a^2b^3)`

Example 3

(a) `(3+sqrt5)^2`

(b) `(sqrta-sqrtb)^2`

(c) `(5+sqrta)(5-sqrta)`

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)(ab) = a2b2

We will use this formula to rationalize denominators.

Example 4

Simplify: `1/(sqrt3-sqrt2`

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.


Q1 `root(5)4root(5)16`

Q2 `sqrta(sqrt(ab)+sqrt(c^3))`

Q3. `(2sqrtx)/(sqrtx-sqrty)`

Q4. `(6sqrta)/(2sqrta-b)`

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