Surds

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1. 1Simplifying surdsWhat a surd is, simplifying √n by its largest square factor, collecting like surds, and multiplying surds and brackets.Done20 min
2. 2Rationalising the denominatorRemoving surds from the bottom of a fraction: multiply by the surd for a single root, and by the conjugate for a sum or difference.Done14 min

Past questions (67)

From 1984–2024: 19 WAEC, 9 NECO, 39 JAMB. Theory questions first, then objective, newest first.

Theory (16)

• NECO 2024 · Paper 2 · Q1Solve the pair of equations $4a + 12b = 16$ and $2a + 3b = 7$. Hence, find the positive value of $K$ given that $a + b = K^2$.
• NECO 2023 · Paper 2 · Q6If $135_k = 231_{\text{four}}$, find the value of $k$.
• NECO 2022 · Paper 2 · Q1If $\sqrt{3} = 1.73$, rationalise $\frac{5}{\sqrt{3}}$ correct to 2 significant figures.
• WAEC 2022 · Paper 2 · Q5Given that $m = \tan30^\circ$ and $n = \tan45^\circ$, simplify, without using a calculator, $\dfrac{m - n}{mn}$, leaving the answer in the form …
• WAEC 2020 · Paper 2 · Q12In the diagram, $\overline{AB}$ is a tangent to the circle with centre $O$ and $COEB$ is a straight line. If $\overline{CD} \parallel \overline{AB}$ …
• WAEC 2020 · Paper 2 · Q3In the diagram, $O$ is the centre of the circle $ABCDE$, $BE$ and $AD$ are diameters, $|BC| = |CD|$ and $\angle BCD = 108^\circ$. Find $\angle CDE$.
• WAEC 2020 · Paper 2 · Q2Given that $\cos60^\circ = \sin30^\circ = \frac12$ and $\cos30^\circ = \sin60^\circ = \frac{\sqrt3}{2}$, evaluate $\dfrac{\tan60^\circ - 1}{1 - \tan30^\circ}$.
• WAEC 2019 · Paper 2 · Q1Given that $110_x = 40_{\text{five}}$, find the value of $x$.
• WAEC 2019 · Paper 2 · Q3Without using mathematical tables or calculators, simplify $3\sqrt{\frac32} - 4\sqrt{\frac23} - \sqrt{24}$.
• WAEC 2018 · Paper 2 · Q2In a right-angled triangle, $\sin X = \frac35$. Evaluate, leaving the answer as a fraction, $5(\cos X)^2 - 3$.
• WAEC 2017 · Paper 2 · Q1Simplify: $\dfrac{2\frac12 + 1\frac34 \div 1\frac25}{2\frac14 - 1\frac12}$.
• WAEC 2014 · Paper 2 · Q2Simplify $3\sqrt{75} - \sqrt{12} + \sqrt{108}$, leaving the answer in surd form (radicals).
• WAEC 2013 · Paper 2 · Q10A segment of a circle is cut off from a rectangular board $22\text{ cm}$ by $12\text{ cm}$ as shown in the diagram, leaving $5\text{ cm}$ and …
• WAEC 2012 · Paper 2 · Q1Simplify: $\dfrac{1\frac14 + \frac79}{1\frac49 - 2\frac23 \times \frac{9}{64}}$.
• WAEC 2009 · Paper 2 · Q1Given that $(\sqrt3 - 5\sqrt2)(\sqrt3 + \sqrt2) = a + b\sqrt6$, find $a$ and $b$.
• WAEC 2009 · Paper 2 · Q1Simplify, without using tables or a calculator, $3\frac13 \div 1\frac14 - \frac49$.

Objective (51)

• NECO 2024 · Paper 1 · Q4Simplify $(4\sqrt3 - 6)(4\sqrt3 + 6)$.
• NECO 2024 · Paper 1 · Q48Calculate the value of $y$ in the diagram, leaving your answer in surd form.
• NECO 2024 · Paper 1 · Q9Rationalize $\dfrac{5}{\sqrt2 + \sqrt3}$.
• WAEC 2024 · Paper 1 · Q6If $(3 - 4\sqrt2)(1 + 3\sqrt2) = a + b\sqrt2$, find the value of $b$.
• NECO 2023 · Paper 1 · Q4Simplify $\dfrac{2\sqrt5}{\sqrt{10}}$.
• WAEC 2023 · Paper 1 · Q47Simplify $3\sqrt{12} + 10\sqrt3 - \dfrac{6}{\sqrt3}$.
• NECO 2022 · Paper 1 · Q16Simplify $\dfrac{3}{\sqrt{3} + 6}$.
• NECO 2022 · Paper 1 · Q5Simplify $2\sqrt{18} - 4\sqrt{32} + 3\sqrt{50}$.
• WAEC 2022 · Paper 1 · Q4If $\sqrt{24} + \sqrt{96} - \sqrt{600} = y\sqrt6$, find the value of $y$.
• WAEC 2021 · Paper 1 · Q12Simplify $2\sqrt7 - \dfrac{14}{\sqrt7} + \dfrac{7}{\sqrt{21}}$.
• WAEC 2020 · Paper 1 · Q8Given that $\dfrac{\sqrt3 + \sqrt5}{\sqrt5} = x + y\sqrt{15}$, find the value of $(x + y)$.
• WAEC 2019 · Paper 1 · Q10Evaluate $2\sqrt{28} - 3\sqrt{50} + \sqrt{72}$.
• JAMB 2018 · UTME · Q22Simplify without using tables $\dfrac{2\sqrt{14} \times 3\sqrt{21}}{7\sqrt{24} \times 2\sqrt{98}}$.
• JAMB 2018 · UTME · Q3Simplify $\dfrac{2\sqrt2 - \sqrt3}{\sqrt2 + \sqrt3}$.
• JAMB 2017 · UTME · Q27Rationalize $\dfrac{2 - \sqrt5}{3 - \sqrt5}$.
• JAMB 2017 · UTME (set 2) · Q18Simplify $4\sqrt{27} + 5\sqrt{12} - 3\sqrt{75}$.
• JAMB 2017 · UTME (set 2) · Q33If $\dfrac{2\sqrt3 - \sqrt2}{\sqrt3 + 2\sqrt2} = m + n\sqrt6$, find the values of $m$ and $n$ respectively.
• JAMB 2016 · UTME · Q21Rationalize $\dfrac{2\sqrt3 + \sqrt5}{\sqrt5 - \sqrt3}$.
• JAMB 2016 · UTME · Q35Rationalize $\dfrac{5\sqrt7 - 7\sqrt5}{\sqrt7 - \sqrt5}$.
• JAMB 2015 · UTME · Q2Find a square root of $170 - 20\sqrt{30}$.
• JAMB 2014 · UTME · Q20What is the common ratio of the G.P. $(\sqrt{10} + \sqrt5), (\sqrt{10} + 2\sqrt5), \ldots$?
• JAMB 2014 · UTME · Q8Simplify $\dfrac{2\sqrt2 - \sqrt3}{\sqrt2 + \sqrt3}$.
• JAMB 2013 · UTME · Q8Simplify $\dfrac{\sqrt5(\sqrt{147} - \sqrt{12})}{\sqrt{15}}$.
• JAMB 2012 · UTME · Q8Simplify $(\sqrt6 + 2)^2 - (\sqrt6 - 2)^2$.
• JAMB 2011 · UTME · Q8Rationalize $\dfrac{2 - \sqrt5}{3 - \sqrt5}$.
• JAMB 2011 · UTME · Q9Simplify $\left(\sqrt2 + \dfrac{1}{\sqrt3}\right)\left(\sqrt2 - \dfrac{1}{\sqrt3}\right)$.
• JAMB 2010 · UTME · Q9Rationalise $\dfrac{2\sqrt3 + \sqrt5}{\sqrt5 - \sqrt3}$.
• JAMB 2004 · UME · Q7Simplify $\dfrac{1}{\sqrt3 + 2}$ in the form $a + b\sqrt3$.
• JAMB 2003 · UME · Q5Simplify $\dfrac{\sqrt{98} - \sqrt{50}}{\sqrt{32}}$.
• JAMB 2002 · UME · Q2Simplify $(\sqrt{0.7} + \sqrt{70})^2$.
• JAMB 2001 · UME · Q5Given that $p = 1 + \sqrt2$ and $q = 1 - \sqrt2$, evaluate $\dfrac{p^2 - q^2}{2pq}$.
• JAMB 2000 · UME · Q3If $\dfrac{2\sqrt3 - \sqrt2}{\sqrt3 + 2\sqrt2} = m + n\sqrt6$, find the values of $m$ and $n$ respectively.
• JAMB 1999 · UME · Q2Find the value of $x$ if $\dfrac{\sqrt2}{x + \sqrt2} = \dfrac{1}{x - \sqrt2}$.
• JAMB 1999 · UME · Q7Simplify $\sqrt{\dfrac{0.0023 \times 750}{0.00345 \times 1.25}}$.
• JAMB 1997 · UME · Q7Simplify $\dfrac{2\sqrt3 + 3\sqrt5}{3\sqrt5 - 2\sqrt3}$.
• JAMB 1995 · UME · Q8Simplify $\dfrac{\sqrt{12} - \sqrt3}{\sqrt{12} + \sqrt3}$.
• JAMB 1994 · UME · Q5Simplify $\sqrt{48} - \dfrac{9}{\sqrt3} + \sqrt{75}$.
• JAMB 1994 · UME · Q6Given that $\sqrt2 = 1.414$, find without using tables the value of $\frac{1}{\sqrt2}$.
• JAMB 1992 · UME · Q6Simplify $5\sqrt{18} - 3\sqrt{72} + 4\sqrt{50}$.
• JAMB 1992 · UME · Q7If $x = 3 - \sqrt3$, find $x^2 + \dfrac{36}{x^2}$.
• JAMB 1991 · UME · Q10Rationalize $\dfrac{2\sqrt3 + 3\sqrt2}{3\sqrt2 - 2\sqrt3}$.
• JAMB 1991 · UME · Q11Simplify $\dfrac{1}{3 + \sqrt5} - \dfrac{1}{3 - \sqrt5}$.
• JAMB 1991 · UME · Q24Simplify $\dfrac{\sqrt{1 + x} + \sqrt x}{\sqrt{1 + x} - \sqrt x}$.
• JAMB 1990 · UME · Q26Given that $x^2 + y^2 + z^2 = 194$, calculate $z$ if $x = 7$ and $\sqrt y = 3$.
• JAMB 1990 · UME · Q8Simplify $\sqrt{160r^2 + \sqrt{71r^4 + \sqrt{100r^8}}}$.
• JAMB 1990 · UME · Q9Simplify $\sqrt{27} + \dfrac{3}{\sqrt3}$.
• JAMB 1989 · UME · Q10Simplify $4 - \dfrac{1}{2 - \sqrt3}$.
• JAMB 1987 · UME · Q15Simplify without using tables $\dfrac{2\sqrt{14} \times 3\sqrt{21}}{7\sqrt{24} \times 2\sqrt{98}}$.
• JAMB 1986 · UME · Q13Simplify $\left(\frac{1}{\sqrt5 + \sqrt3} - \frac{1}{\sqrt5 - \sqrt3}\right) \times \frac{1}{\sqrt3}$.
• JAMB 1986 · UME · Q21Solve the equation $3x^2 + 6x - 2 = 0$.
• JAMB 1984 · UME · Q13Rationalize $\dfrac{5\sqrt7 - 7\sqrt5}{\sqrt7 - \sqrt5}$.