# Multiplication of radicals using the distributive property

Learn how to perform multiplication of radicals with some carefully chosen examples.

Example #1

 $$(\sqrt{3} + \sqrt{2}) × (\sqrt{3} - \sqrt{2})$$

Here is how to use the distributive property to do this multiplication.

 $$(\sqrt{3} + \sqrt{2}) × (\sqrt{3} - \sqrt{2})$$
 $$= \sqrt{3}×\sqrt{3} - \sqrt{3}×\sqrt{2} + \sqrt{2}×\sqrt{3} - \sqrt{2}×\sqrt{2}$$
 $$= \sqrt{3}×\sqrt{3} - \sqrt{3}×\sqrt{2} + \sqrt{2}×\sqrt{3}$$
 $$- \sqrt{2}×\sqrt{2}$$
 $$= \sqrt{3 × 3} - \sqrt{3 × 2} + \sqrt{2 × 3} - \sqrt{2 × 2}$$
 $$= \sqrt{9} - \sqrt{6} + \sqrt{6} - \sqrt{4}$$
 $$= \sqrt{9} - \sqrt{4}$$

= 3 - 2

=  1

Did you notice that the multiplication has the format (a + b) x (a - b)?

(a + b) x (a - b) = a2 - b2

Therefore, next time, there is no need to do all this math above.

 $$(\sqrt{3} + \sqrt{2}) × (\sqrt{3} - \sqrt{2}) = (\sqrt{3})^2 - (\sqrt{2})^2$$
 $$Notice \ also \ that (\sqrt{3})^2 = 3$$
 $$In \ general,\sqrt{a} ×\sqrt{a} = (\sqrt{a})^2 = a$$

Example #2

 $$(\sqrt{7} + \sqrt{5}) × (\sqrt{7} - \sqrt{5})$$
 $$= (\sqrt{7})^2 - (\sqrt{5})^2 = 7 - 5 = 2$$

Example #3

 $$(\sqrt{6} + \sqrt{9}) × (\sqrt{6} - \sqrt{9}) = 6 - 9 = -3$$

Example #4

 $$(\sqrt{4} - 2\sqrt{12}) × (\sqrt{4} + \sqrt{12})$$
 $$= \sqrt{4} × \sqrt{4} + \sqrt{4} × \sqrt{12} -2\sqrt{12} ×\sqrt{4} -2 \sqrt{12} × \sqrt{12}$$
 $$= (\sqrt{4} × \sqrt{4} + \sqrt{4} × \sqrt{12} -2\sqrt{12} ×\sqrt{4}$$
 $$-2 \sqrt{12} × \sqrt{12}$$
 $$= 4 + 2 × \sqrt{12} -2\sqrt{12} ×2 -2 × 12$$
 $$= 4 + 2\sqrt{12} -4\sqrt{12} -24$$
 $$= -2\sqrt{12} - 20$$

## Recent Articles

1. ### How to estimate fractions

Dec 11, 17 08:45 AM

Learn how to estimate fractions with this easy to follow lesson.

New math lessons

Your email is safe with us. We will only use it to inform you about new math lessons.

## Recent Lessons

1. ### How to estimate fractions

Dec 11, 17 08:45 AM

Learn how to estimate fractions with this easy to follow lesson.

Tough Algebra Word Problems.

If you can solve these problems with no help, you must be a genius!

Everything you need to prepare for an important exam!

K-12 tests, GED math test, basic math tests, geometry tests, algebra tests.

Real Life Math Skills

Learn about investing money, budgeting your money, paying taxes, mortgage loans, and even the math involved in playing baseball.