E1.18 Surds

Learning objectives

E1.18.1Understand and use surds, including simplifying expressions.• Understand and use surds, including simplifying expressions.E1.18.2Rationalise the denominator• Rationalise the denominator. Examples include: • 20 25= • 200 32 62− =. Examples include: • 10 25= • − + = +.

A surd is an irrational root kept in exact form, such as $\sqrt3$ . Simplify it by extracting square factors, then combine only terms with the same remaining surd.

$\sqrt{ab}=\sqrt a\sqrt b,\qquad \sqrt a\sqrt a=a\quad(a,b\ge0)$

Choose the largest square factor. Since $300=100\times3$ and $48=16\times3$ , $\sqrt{300}+\sqrt{48}=10\sqrt3+4\sqrt3=14\sqrt3$ . The matching $\sqrt3$ terms combine like algebraic terms.

Operation	Safe move	Example
add or subtract	simplify first, then combine like surds	$5\sqrt2-3\sqrt2=2\sqrt2$
multiply	multiply coefficients and radicands	$2\sqrt3\times4\sqrt6=8\sqrt{18}=24\sqrt2$
expand brackets	use ordinary distributive multiplication	$\sqrt5\times\sqrt5=5$

For $(3-\sqrt5)(2+3\sqrt5)$ , expand to $6+9\sqrt5-2\sqrt5-15$ . Collect rational and surd terms separately to obtain $-9+7\sqrt5$ .

Do not add unlike surds or split a root across addition: $\sqrt2+\sqrt3$ cannot be simplified, and $\sqrt{a+b}$ is not generally $\sqrt a+\sqrt b$ . Extract only genuine square factors and leave the answer exact.

To rationalise a denominator, multiply the fraction by a form of 1 that removes every surd from the denominator while preserving the fraction's value.

Denominator	Multiply numerator and denominator by	Why it works
$\sqrt c$	$\sqrt c$	$\sqrt c\times\sqrt c=c$
$a+\sqrt c$	$a-\sqrt c$	conjugates give $a^2-c$
$a-\sqrt c$	$a+\sqrt c$	conjugates give $a^2-c$

For a single surd, $\dfrac{6}{\sqrt{10}}\times\dfrac{\sqrt{10}}{\sqrt{10}}=\dfrac{6\sqrt{10}}{10}=\dfrac{3\sqrt{10}}5$ . Simplify the numerical fraction after the denominator becomes rational.

$(a+b\sqrt c)(a-b\sqrt c)=a^2-b^2c$

For $\dfrac1{\sqrt7+2}$ , use the conjugate $\sqrt7-2$ : $\dfrac1{\sqrt7+2}\times\dfrac{\sqrt7-2}{\sqrt7-2}=\dfrac{\sqrt7-2}{7-4}=\dfrac{\sqrt7-2}{3}$ .

Multiplying only the denominator changes the value; multiply the numerator by the same non-zero expression. For a two-term denominator, using the same sign creates another surd term, so use the conjugate with the opposite sign and simplify fully.

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—Cambridge IGCSE Mathematics (0580)Read

E1.18 Surds

Learning objectives

Simplify and calculate with surds

Rationalise surd denominators

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