7.6 Surds and exact calculations — Learn

Prerequisites: square numbers, factors and simplifying fractions.

Before you begin

You should be confident with square numbers, factors and simplifying fractions.

Objectives

• Simplify surds by taking square factors outside the root.
• Add and subtract like surds.
• Multiply and divide surds.
• Rationalise a denominator in a simple exact calculation.

Simplifying surds

A surd is an exact root that cannot be written as a whole number. Split a number under a square root into a square factor and the remaining factor.

Example 1. Simplify $\sqrt{72}$.

Example 2. Simplify $3\sqrt8+\sqrt{18}$.

Only like surds can be added, as with matching variables.

Products and denominators

Multiply surds by multiplying the numbers under the roots. Rationalising a denominator rewrites a fraction so its denominator has no root.

Example 3. Simplify $\sqrt6\times\sqrt{15}$.

Example 4. Rationalise $\frac{5}{\sqrt3}$.

Watch out. A common error is to write $\sqrt a+\sqrt b=\sqrt{a+b}$. For example, $\sqrt9+\sqrt{16}=7$, but $\sqrt{25}=5$. Simplify each root first, then combine only matching surds.

Progress check. Before simplifying $\sqrt{50}+\sqrt8$, identify the square factor in each radicand.

You should now be able to

• simplify square roots by taking out square factors;
• combine and multiply surds exactly;
• rationalise a simple denominator.