A surd is an irrational root written in exact form. sqrt(ab)= a b (a,b 0) sqrt(ab)= a b (a 0, b>0) Simplifying surds **Operation** | **Example** Addition and subtraction | 4 5+3 5=7 5 2 3+5 7: unlike surds, cannot combine Multiplication | 3 2 x 5 2=15(2)=30 2 3 x 4 5=8sqrt(15) Division | 9sqrt(22)3sqrt(11)=3sqrt(2) Expanding brackets **Identity** | **Example** (A+B)2=A2+2AB+B2 | (3 2+1)2=(3 2)2+2(3 2)+12 =18+6 2+1 =19+6 2 (A+B)(A-B)=A2-B2 | (2 5+3)(2 5-3)=(2 5)2-32 =20-9 =11 Conjugates have opposite signs between their terms: a+ b and a- b. Simplest and entire forms **Task** | **Method** | **Example** Simplest form a b | Take out the largest square factor of the radicand. | sqrt(180)=sqrt(36 x 5) =6 5 Entire surd p | Bring the coefficient under the root sign. | 5 2=sqrt(25) x 2 =sqrt(50) Roots do not split over sums or differences: sqrt(a+b)!= a+ b in general.

A surd is an irrational root written in exact form.

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

\(\sqrt{\dfrac ab}=\dfrac{\sqrt a}{\sqrt b}\quad(a\ge0,\ b>0)\)

\(4\sqrt5+3\sqrt5=7\sqrt5\)
\(2\sqrt3+5\sqrt7\): unlike surds, cannot combine

\(\begin{aligned} 3\sqrt2\times5\sqrt2&=15(2)=30 \\ 2\sqrt3\times4\sqrt5&=8\sqrt{15} \end{aligned}\)

\(\dfrac{9\sqrt{22}}{3\sqrt{11}}=3\sqrt{2}\)

\(\begin{aligned} (3\sqrt2+1)^2&=(3\sqrt2)^2+2(3\sqrt2)+1^2 \\ &=18+6\sqrt2+1 \\ &=19+6\sqrt2 \end{aligned}\)

\(\begin{aligned} (2\sqrt5+3)(2\sqrt5-3)&=(2\sqrt5)^2-3^2 \\ &=20-9 \\ &=11 \end{aligned}\)

Conjugates have opposite signs between their terms: \(a+\sqrt b\) and \(a-\sqrt b\).

Take out the largest square factor of the radicand.

\(\begin{aligned} \sqrt{180}&=\sqrt{36\times5} \\ &=6\sqrt5 \end{aligned}\)

Bring the coefficient under the root sign.

\(\begin{aligned} 5\sqrt2&=\sqrt{25}\times\sqrt2 \\ &=\sqrt{50} \end{aligned}\)

Roots do not split over sums or differences: \(\sqrt{a+b}\ne\sqrt a+\sqrt b\) in general.

Law of Surds

Law of Surds