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, .

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

Question

Simplify the following surds:

\((\sqrt{7})^2\)
\((\sqrt{11})^2\)
\((\sqrt{15})^2\)

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

Simplify \(\sqrt{18} \times \sqrt{2}\)
Simplify \(\frac{\sqrt{88}}{2}\)
Multiply out \(\sqrt{11}(2 - \sqrt{3})\)

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Surds - AQAMultiplying and dividing surds