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CDS Mathematics · Formula sheet

Surds, Indices and Simplification formulas

18 formulas and 18 common traps for CDS Mathematics Surds, Indices and Simplification, grouped by subtopic.

Full notes with worked examples

Fractions and Decimals

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Recurring decimals as fractions

Mixed recurring decimal

0.ab‾=ab‾−a900.a\overline{b} = \dfrac{\overline{ab} - a}{90}

Comparing fractions and fractions of a whole

Cross-multiplication

ab>cd  ⟺  ad>bc(b,d>0)\dfrac ab > \dfrac cd \iff ad > bc \quad (b, d > 0)

Simplifying decimal expressions

Square of a sum

a2+2ab+b2=(a+b)2a^2 + 2ab + b^2 = (a + b)^2

Common traps

Read where the bar starts

0.53‾0.5\overline{3} repeats only the 33 (=4890= \dfrac{48}{90}); 0.53‾0.\overline{53} repeats both digits (=5399= \dfrac{53}{99}). The same four symbols give two different numbers.

The remainder, not the total

'1120\dfrac{11}{20} of those who appeared' is a fraction of the students left after the absentees, not of everyone registered.

Count decimal places when you take a root

0.0081=0.09\sqrt{0.0081} = 0.09, not 0.90.9: the root has half as many decimal places as the number.

Laws of Indices

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The index laws

Index laws

aman=am+n,(am)n=amn,a−n=1ana^m a^n = a^{m + n}, \quad (a^m)^n = a^{mn}, \quad a^{-n} = \dfrac{1}{a^n}

When several powers are equal

Common value k

ax=by=cz=k  ⇒  a=k1/x, b=k1/y, c=k1/za^x = b^y = c^z = k \;\Rightarrow\; a = k^{1/x},\ b = k^{1/y},\ c = k^{1/z}

Comparing powers and roots

Common exponent

amn=(am)na^{mn} = (a^m)^n

Common traps

The base must not be 1

xp=xqx^{p} = x^{q} gives p=qp = q only if x≠1x \ne 1 (and x>0x > 0). That is why the stems add 'x≠1x \ne 1', and why x=1x = 1 is often an extra solution of an index equation.

Reciprocals of the exponents, not the exponents

10=2×510 = 2\times 5 gives 1z=1x+1y\dfrac1z = \dfrac1x + \dfrac1y, not z=x+yz = x + y. Products of the bases become SUMS of the reciprocal exponents.

Raise every number to the same power

Comparing 2\sqrt2 and 33\sqrt[3]3 by squaring one and cubing the other proves nothing. Use one exponent, the LCM of the root indices, for all of them.

Exponential Equations

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Substitute t = a^x

Quadratic in t

a1+x+a1−x=c  ⇒  at2−ct+a=0,  t=axa^{1 + x} + a^{1 - x} = c \;\Rightarrow\; a t^2 - c t + a = 0,\ \ t = a^x

Take out the common power

Common factor

ax+1−ax−1=ax−1(a2−1)a^{x + 1} - a^{x - 1} = a^{x - 1}(a^2 - 1)

Common traps

Reject non-positive t

A power axa^x with a>0a > 0 is always positive. A root t=−2t = -2 of the quadratic gives no xx, and counting it doubles the answer.

Answer what is asked

Solving for xx is usually only the first step: the question may want 102x10^{2x} or x+yx + y. Keep the requested expression in view.

Square Roots of Surds

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The square root of a + 2√b

Square root of a surd

a+2b=m+n,m+n=a, mn=b\sqrt{a + 2\sqrt b} = \sqrt m + \sqrt n, \quad m + n = a,\ mn = b

A surd plus its reciprocal

Conjugates with product 1

(2+3)(2−3)=1(2 + \sqrt3)(2 - \sqrt3) = 1

Common traps

The principal root is positive

9−45\sqrt{9 - 4\sqrt5} is 5−2\sqrt5 - 2, not 2−52 - \sqrt5: both square to the same number, but only the first is positive. One CDS paper printed only the negative form among its options.

Square root first, then the reciprocal

For x=7+43x = 7 + 4\sqrt3, x+1x=14x + \dfrac1x = 14 but x+1x=4\sqrt x + \dfrac{1}{\sqrt x} = 4. Check whether the question has xx or x\sqrt x before adding.

Rationalising and Conjugate Pairs

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Conjugate pairs x and y

Rationalising

a+ba−b=(a+b)2a−b\dfrac{\sqrt a + \sqrt b}{\sqrt a - \sqrt b} = \dfrac{(\sqrt a + \sqrt b)^2}{a - b}

Telescoping sums

Telescoping term

1n+n+1=n+1−n\dfrac{1}{\sqrt n + \sqrt{n + 1}} = \sqrt{n + 1} - \sqrt n

Common traps

Sign of a difference

a−ba+b\dfrac{\sqrt a - \sqrt b}{\sqrt a + \sqrt b} is the smaller of the two twins, so 'small minus large' is negative. An option with the right size but the wrong sign is always printed.

Find the first and last terms exactly

A sum ending at 1195+196\dfrac{1}{\sqrt{195} + \sqrt{196}} leaves 196\sqrt{196}, not 195\sqrt{195}. Write the first and last terms out before cancelling.

Equations with Surds

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Componendo and dividendo

A+BA−B=k  ⇒  AB=k+1k−1\dfrac{\sqrt A + \sqrt B}{\sqrt A - \sqrt B} = k \;\Rightarrow\; \dfrac{\sqrt A}{\sqrt B} = \dfrac{k + 1}{k - 1}

Factorising with square roots

Difference of squares in roots

x−y=(x−y)(x+y)x - y = (\sqrt x - \sqrt y)(\sqrt x + \sqrt y)

Common traps

k + 1 over k − 1, not the other way

From S+DS−D=k\dfrac{S + D}{S - D} = k the ratio SD\dfrac SD is k+1k−1\dfrac{k + 1}{k - 1}. Swapping it gives the reciprocal, which is usually an option.

Square the ratio at the end

x=2y\sqrt x = 2\sqrt y gives xy=4\dfrac xy = 4, not 22. The ratio of the roots is the square root of the ratio asked for.

Simplifying Surd Expressions

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Expressions built from cube roots

Cube with u³ = a

(u+u2)3=u3(1+u)3=a (1+u)3(u + u^2)^3 = u^3(1 + u)^3 = a\,(1 + u)^3

Square roots of squares and products of sums

Root of a square

a2=∣a∣\sqrt{a^2} = |a|

Common traps

Move the whole number before cubing

Cubing x=2+u+u2x = 2 + u + u^2 directly produces dozens of terms. Cubing x−2=u+u2x - 2 = u + u^2 produces four, because u3u^3 factors out.

√(a²) is not a

(a−b)2=∣a−b∣\sqrt{(a - b)^2} = |a - b|, which is b−ab - a when b>ab > a. Writing a−ba - b makes a positive quantity look negative.

Continued Fractions and Nested Radicals

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Continued fractions

One step of the decomposition

pq=a+1  q/r  ,a=⌊pq⌋\dfrac pq = a + \cfrac{1}{\;q/r\;}, \quad a = \left\lfloor \tfrac pq \right\rfloor

Infinite nested radicals

Self-similar nest

x=a+x  ⇒  x2−x−a=0x = \sqrt{a + x} \;\Rightarrow\; x^2 - x - a = 0

Common traps

Invert before splitting

If the given value is 1a+⋯=1623\dfrac{1}{a + \cdots} = \dfrac{16}{23}, the first whole part comes from 2316\dfrac{23}{16}, not from 1623\dfrac{16}{23} (whose whole part is 00).

Drop the negative root

x2−x−12=0x^2 - x - 12 = 0 has roots 44 and −3-3; a square root is never negative, so the value is 44. Likewise x2=4xx^2 = 4x gives x=4x = 4, not 00.

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