# 5. Multiplication and Division of Radicals

### Later, on this page...

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

`root(n)axxroot(n)b=root(n)(ab)`

### Example 1

(a) `sqrt5sqrt2`

(b) `sqrt33sqrt3`

### 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)(a−b) =a^{2}−b^{2}

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.

### Exercises

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

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

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

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

Didn't find what you are looking for on this page? Try **search**:

### Online Algebra Solver

This algebra solver can solve a wide range of math problems. (Please be patient while it loads.)

Go to: Online algebra solver

### Ready for a break?

Play a math game.

(Well, not really a math game, but each game was made using math...)

### The IntMath Newsletter

Sign up for the free **IntMath Newsletter**. Get math study tips, information, news and updates each fortnight. Join thousands of satisfied students, teachers and parents!

### Share IntMath!

### Algebra Lessons on DVD

Easy to understand algebra lessons on DVD. See samples before you commit.

More info: Algebra videos