For these real square-root laws, the radicands are non-negative; a denominator radicand must be strictly positive. Thus the product law uses \(a,b\ge0\), and the quotient law uses \(a\ge0,b>0\). A coefficient moved inside a principal square root must also be non-negative. From O-Level to Additional Mathematics A surd is an irrational root such as \(\sqrt2\) or \(5\sqrt3\), written exactly with the root sign. You already evaluate square roots on a calculator; Additional Mathematics keeps them as surds because a surd answer is an exact answer — never replace one with a rounded decimal mid-working. Laws of surds OperationLawExample Addition and subtraction\(k\sqrt a\pm h\sqrt a=(k\pm h)\sqrt a\) Like surds only\(4\sqrt5+3\sqrt5=7\sqrt5\) \(2\sqrt3+5\sqrt7\): unlike surds, cannot combine Multiplication\(\sqrt a\times\sqrt a=a\) \(\sqrt a\times\sqrt b=\sqrt{ab}\)\(3\sqrt2\times5\sqrt2=15(2)=30\) \(2\sqrt3\times4\sqrt5=8\sqrt{15}\) Division\(\dfrac{\sqrt a}{\sqrt b}=\sqrt{\dfrac ab}\), \(b>0\)\(\dfrac{9\sqrt{22}}{3\sqrt{11}}=3\sqrt{2}\) Expansion with surds IdentityExample \((A+B)^2=A^2+2AB+B^2\)\((3\sqrt2+1)^2=(3\sqrt2)^2+2(3\sqrt2)+1^2\) \(=18+6\sqrt2+1\) \(=19+6\sqrt2\) \((A+B)(A-B)=A^2-B^2\)\((2\sqrt5+3)(2\sqrt5-3)=(2\sqrt5)^2-3^2\) \(=20-9\) \(=11\) For rational \(p,q,r\) with \(r\ge0\), the expressions \(p+q\sqrt r\) and \(p-q\sqrt r\) are conjugates. Their product is \(p^2-q^2r\), which is rational. Simplest form \(a\sqrt b\) and entire surd \(\sqrt p\) TaskMethodExample Simplest form \(a\sqrt b\)Take out the largest square factor of the radicand.\(\sqrt{180}=\sqrt{36\times5}\) \(=6\sqrt5\) Entire surd \(\sqrt p\)Bring the coefficient under the root sign.\(5\sqrt2=\sqrt{25}\times\sqrt2\) \(=\sqrt{50}\) Take out the largest square factor in one step. Writing \(\sqrt{180}=\sqrt9\times\sqrt{20}=3\sqrt{20}\) is not simplest form, because \(\sqrt{20}\) still contains the square factor \(4\). Exam check Square roots do not distribute over addition or subtraction: \(\sqrt{a+b}=\sqrt a+\sqrt b\) and \(\sqrt{a-b}=\sqrt a-\sqrt b\) are not valid identities in general. For example, \(\sqrt{9+16}=5\), whereas \(3+4=7\). Products and quotients split under the stated real-domain conditions.

Surd Laws and Simplest Form

Secondary 3

TGM Original Questions

For these real square-root laws, the radicands are non-negative; a denominator radicand must be strictly positive. Thus the product law uses \(a,b\ge0\), and the quotient law uses \(a\ge0,b>0\). A coefficient moved inside a principal square root must also be non-negative.

From O-Level to Additional Mathematics

A surd is an irrational root such as \(\sqrt2\) or \(5\sqrt3\), written exactly with the root sign. You already evaluate square roots on a calculator; Additional Mathematics keeps them as surds because a surd answer is an exact answer — never replace one with a rounded decimal mid-working.

Laws of surds

Operation	Law	Example
Addition and subtraction	\(k\sqrt a\pm h\sqrt a=(k\pm h)\sqrt a\) Like surds only	\(4\sqrt5+3\sqrt5=7\sqrt5\) \(2\sqrt3+5\sqrt7\): unlike surds, cannot combine
Multiplication	\(\sqrt a\times\sqrt a=a\) \(\sqrt a\times\sqrt b=\sqrt{ab}\)	\(3\sqrt2\times5\sqrt2=15(2)=30\) \(2\sqrt3\times4\sqrt5=8\sqrt{15}\)
Division	\(\dfrac{\sqrt a}{\sqrt b}=\sqrt{\dfrac ab}\), \(b>0\)	\(\dfrac{9\sqrt{22}}{3\sqrt{11}}=3\sqrt{2}\)

Expansion with surds

Identity	Example
\((A+B)^2=A^2+2AB+B^2\)	\((3\sqrt2+1)^2=(3\sqrt2)^2+2(3\sqrt2)+1^2\) \(=18+6\sqrt2+1\) \(=19+6\sqrt2\)
\((A+B)(A-B)=A^2-B^2\)	\((2\sqrt5+3)(2\sqrt5-3)=(2\sqrt5)^2-3^2\) \(=20-9\) \(=11\)

For rational \(p,q,r\) with \(r\ge0\), the expressions \(p+q\sqrt r\) and \(p-q\sqrt r\) are conjugates. Their product is \(p^2-q^2r\), which is rational.

Simplest form \(a\sqrt b\) and entire surd \(\sqrt p\)

Task	Method	Example
Simplest form \(a\sqrt b\)	Take out the largest square factor of the radicand.	\(\sqrt{180}=\sqrt{36\times5}\) \(=6\sqrt5\)
Entire surd \(\sqrt p\)	Bring the coefficient under the root sign.	\(5\sqrt2=\sqrt{25}\times\sqrt2\) \(=\sqrt{50}\)

Take out the largest square factor in one step. Writing \(\sqrt{180}=\sqrt9\times\sqrt{20}=3\sqrt{20}\) is not simplest form, because \(\sqrt{20}\) still contains the square factor \(4\).

Exam check

Square roots do not distribute over addition or subtraction: \(\sqrt{a+b}=\sqrt a+\sqrt b\) and \(\sqrt{a-b}=\sqrt a-\sqrt b\) are not valid identities in general. For example, \(\sqrt{9+16}=5\), whereas \(3+4=7\). Products and quotients split under the stated real-domain conditions.

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