Chapter Contents

• Exponents and Radicals
• 1. Simplifying Expressions with Integral Exponents
• 2. Fractional Exponents
• 3. Simplest Radical Form
• 4. Addition and Subtraction of Radicals
• 5. Multiplication and Division of Radicals (Rationalizing the Denominator)
• 6. Equations with Radicals
• Exponents and Radicals Problem Solver

• Comments, Questions?

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5. Multiplication and Division of Radicals

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`

Answer

(b) `sqrt33sqrt3`

Answer

Example 2

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

Answer

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

Answer

Example 3

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

Answer

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

Answer

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

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 4

Simplify: `1/(sqrt3-sqrt2`

Answer

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`

Answer

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

Answer

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

Answer

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

Answer