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Surds can be simplified if the number in the surd has a square number as a factor.

Remember these general rules:

• \(\sqrt{ab} = \sqrt{a} \times \sqrt{b}\)
• \(\sqrt{a} \times \sqrt{a} = a\)
• \(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{a \div b}\)

Example

Simplify \(\sqrt{12}\)

\(12 = 3 \times 4\), so we can write \(\sqrt{12} = \sqrt{(4 \times 3)} = \sqrt{4} \times \sqrt{3}\)

\(\sqrt{4} = 2\) so \(\sqrt{12} = 2\sqrt{3}\)

Simplify \(\sqrt{10} \times \sqrt{5}\)

\(\sqrt{10} \times \sqrt{5} = \sqrt{50} \)

\(50 = 25 \times 2\), so we can write \(\sqrt{50} = \sqrt{(25 \times 2)} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2}\)

Simplify \(\frac{\sqrt{12}}{\sqrt{6}}\)

\(\frac{\sqrt{12}}{\sqrt{6}} = \sqrt{(12 \div 6)} = \sqrt{2}\)

Example

\(5 \sqrt{2} - 3 \sqrt{2} = 2 \sqrt{2} \)

This is just like collecting like terms in an expression
\(4 \sqrt{2} - 3 \sqrt{3}\) cannot be added since the numbers inside the square roots are not the same.

It may be necessary to simplify one or more surds in an expression first, before adding or subtracting the surds.

Example

\(\sqrt{12} + \sqrt{27}\)

\(12 = 3 \times 4\) so \( \sqrt{12} = \sqrt{(3 \times 4)} = 2 \sqrt{3}\)

\(27 = 3 \times 9\) so \(\sqrt{27} = \sqrt{(3 \times 9)} = 3 \sqrt{3}\)

\(\sqrt{12} + \sqrt{27} = 2 \sqrt{3} + 3 \sqrt{3} = 5 \sqrt{3}\)

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 with the same number inside the square root gives a whole, rational number.

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

Multiplying surds with different numbers inside the square root

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

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

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

\(2 \sqrt{3} \times 3 \sqrt{2} = \sqrt{(4 \times 3)} \times \sqrt{(9 \times 2)} = \sqrt{12} \times \sqrt{18} = \sqrt{216} = \sqrt{(36 \times 6)} = 6 \sqrt{6}\)

The quicker way of doing this is by multiplying the component parts:

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

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

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

Divide the whole numbers:

\(8 \div 2 = 4\)

Divide the square roots:

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

\(4 \sqrt{2}\)

Expressions with brackets that include surds can be multiplied out in a similar way to multiplying brackets to give a quadratic expression.

Simplify fully \((3 + \sqrt{2})(2 + \sqrt{5})\)

Each term in the first bracket has to be multiplied by each term in the second bracket. One way to do this is to use a grid:

The four terms cannot be simplified because each of the surds has a different number inside the square root, and none of the surds can be simplified.

\((3 + \sqrt{2})(2 + \sqrt{5}) = 6 + 2 \sqrt{2} + 3 \sqrt{5} + \sqrt{10}\)

The same method can be used if the numbers in the surds are the same:

Simplify fully \((1 + \sqrt{2})(5 + \sqrt{5})\)

The surds have the same number inside the square root, so they give a rational number when multiplied together. The four terms can be simplified by adding together the rational terms and the irrational terms:

\((5) + (-3) = 2\) and \((5 \sqrt{3}) + (- \sqrt{3}) = 4 \sqrt{3}\), so \((1 + \sqrt{3})(5 - \sqrt{3}) = 2 + 4 \sqrt{3}\)

Example

Simplify \(\frac{\sqrt{8}}{\sqrt{6}}\)

The denominator can be rationalised by multiplying the numerator and denominator by √6.

\(\frac{\sqrt{8} \times \sqrt{6}}{\sqrt{6} \times \sqrt{6}} = \frac{\sqrt{48}}{6} = \frac{\sqrt{(16 \times 3)}}{6} = \frac{4 \sqrt{3}}{6} = \frac{2 \sqrt{3}}{3}\)