Multiplying and dividing surds

Multiplying surds with the same number inside the square root

We know that:

\sqrt{2} \times \sqrt{2} = 2

\sqrt{5} \times \sqrt{5} = 5

So multiplying surds that have the same number inside the square root gives a whole, rational number.

(\sqrt{3})^2 = \sqrt{3} \times \sqrt{3} = \sqrt{9} = 3

Question

Simplify the following surds:

  1. (\sqrt{7})^2
  2. (\sqrt{11})^2
  3. (\sqrt{15})^2
  1. 7
  2. 11
  3. 15

Multiplying surds with different numbers inside the square root

First, simplify the numbers inside the square roots if possible, then multiply them.

Examples

1. \sqrt{8} \times \sqrt{10} = \sqrt{80}

\sqrt{8} = \sqrt{4 \times 2}

= \sqrt{4} \times \sqrt{2}

=  2\sqrt{2}

\sqrt{10} = \sqrt{2} \times \sqrt{5}

\sqrt{8} \times \sqrt{10} = 2\sqrt{2} \times \sqrt{2} \times \sqrt{5}

= 2 \times 2 \times \sqrt{5}

= 4\sqrt{5}

2. Multiply  2\sqrt{3} \times 3\sqrt{2}

First multiply the whole numbers:

2 \times 3 = 6

Then multiply the surds:

\sqrt{3} \times \sqrt{2} = \sqrt6

This makes: 6\sqrt{6}

Dividing surds

Just like the method used to multiply, the quicker way of dividing is by dividing the component parts:

\frac{8 \sqrt{6}}{2 \sqrt{3}}

Divide the whole numbers:

8 \div 2 = 4

Divide the square roots:

\frac{\sqrt{6}}{\sqrt{3}} = \sqrt{2}

So the answer is:

4 \sqrt{2}

Question
  1. Simplify \sqrt{18} \times \sqrt{2}
  2. Simplify \frac{\sqrt{88}}{2}
  3. Multiply out \sqrt{11}(2 - \sqrt{3})
  1. 6
  2. \sqrt{22}
  3. 2\sqrt{11} - \sqrt{33}