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

\(\sqrt{8} \times \sqrt{10} = \sqrt{80}\)

\(\sqrt{80} = \sqrt{(16 \times 5)} = 4 \times \sqrt{5} = 4 \sqrt{5}\)

Example

\(2 \sqrt{3} \times 3 \sqrt{2}\)

Multiply the whole numbers:

\(2 \times 3 = 6\)

Multiply the surds:

\(\sqrt{3} \times \sqrt{2} = \sqrt{6}\)

This makes: \(6 \sqrt{6}\)

Dividing surds

Just like with multiplication, deal with the component parts separately.

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

Divide the whole numbers:

\(8 \div 2 = 4\)

Divide the square roots:

\(\sqrt{6} \div \sqrt{3} = \sqrt{2}\)

\(4 \sqrt{2}\)

Question

Simplify the following surds:

\(\sqrt{18} \times \sqrt{2}\)
\(\frac{\sqrt{88}}{2}\)
\(\frac{14\sqrt{30}}{16\sqrt{3}}\)

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Surds - Higher - OCRMultiplying and dividing surds