Surds and Indices — CDS Elementary Mathematics

Formulas (3)

• Recurring decimals as fractions · Mixed recurring decimal
  $$0.a\overline{b} = \dfrac{\overline{ab} - a}{90}$$
• Comparing fractions and fractions of a whole · Cross-multiplication
  $$\dfrac ab > \dfrac cd \iff ad > bc \quad (b, d > 0)$$
• Simplifying decimal expressions · Square of a sum
  $$a^2 + 2ab + b^2 = (a + b)^2$$

Watch out for (3)

• Read where the bar starts
  → Recurring decimals as fractions
• The remainder, not the total
  → Comparing fractions and fractions of a whole
• Count decimal places when you take a root
  → Simplifying decimal expressions

Formulas (3)

• The index laws · Index laws
  $$a^m a^n = a^{m + n}, \quad (a^m)^n = a^{mn}, \quad a^{-n} = \dfrac{1}{a^n}$$
• When several powers are equal · Common value k
  $$a^x = b^y = c^z = k \;\Rightarrow\; a = k^{1/x},\ b = k^{1/y},\ c = k^{1/z}$$
• Comparing powers and roots · Common exponent
  $$a^{mn} = (a^m)^n$$

Watch out for (3)

• The base must not be 1
  → The index laws
• Reciprocals of the exponents, not the exponents
  → When several powers are equal
• Raise every number to the same power
  → Comparing powers and roots

Formulas (2)

• Substitute t = a^x · Quadratic in t
  $$a^{1 + x} + a^{1 - x} = c \;\Rightarrow\; a t^2 - c t + a = 0,\ \ t = a^x$$
• Take out the common power · Common factor
  $$a^{x + 1} - a^{x - 1} = a^{x - 1}(a^2 - 1)$$

Watch out for (2)

• Reject non-positive t
  → Substitute t = a^x
• Answer what is asked
  → Take out the common power

Formulas (2)

• The square root of a + 2√b · Square root of a surd
  $$\sqrt{a + 2\sqrt b} = \sqrt m + \sqrt n, \quad m + n = a,\ mn = b$$
• A surd plus its reciprocal · Conjugates with product 1
  $$(2 + \sqrt3)(2 - \sqrt3) = 1$$

Watch out for (2)

• The principal root is positive
  → The square root of a + 2√b
• Square root first, then the reciprocal
  → A surd plus its reciprocal

Formulas (2)

• Conjugate pairs x and y · Rationalising
  $$\dfrac{\sqrt a + \sqrt b}{\sqrt a - \sqrt b} = \dfrac{(\sqrt a + \sqrt b)^2}{a - b}$$
• Telescoping sums · Telescoping term
  $$\dfrac{1}{\sqrt n + \sqrt{n + 1}} = \sqrt{n + 1} - \sqrt n$$

Watch out for (2)

• Sign of a difference
  → Conjugate pairs x and y
• Find the first and last terms exactly
  → Telescoping sums

Formulas (2)

• Componendo and dividendo · Componendo and dividendo
  $$\dfrac{\sqrt A + \sqrt B}{\sqrt A - \sqrt B} = k \;\Rightarrow\; \dfrac{\sqrt A}{\sqrt B} = \dfrac{k + 1}{k - 1}$$
• Factorising with square roots · Difference of squares in roots
  $$x - y = (\sqrt x - \sqrt y)(\sqrt x + \sqrt y)$$

Watch out for (2)

• k + 1 over k − 1, not the other way
  → Componendo and dividendo
• Square the ratio at the end
  → Factorising with square roots

Formulas (2)

• Expressions built from cube roots · Cube with u³ = a
  $$(u + u^2)^3 = u^3(1 + u)^3 = a\,(1 + u)^3$$
• Square roots of squares and products of sums · Root of a square
  $$\sqrt{a^2} = |a|$$

Watch out for (2)

• Move the whole number before cubing
  → Expressions built from cube roots
• √(a²) is not a
  → Square roots of squares and products of sums

Formulas (2)

• Continued fractions · One step of the decomposition
  $$\dfrac pq = a + \cfrac{1}{\;q/r\;}, \quad a = \left\lfloor \tfrac pq \right\rfloor$$
• Infinite nested radicals · Self-similar nest
  $$x = \sqrt{a + x} \;\Rightarrow\; x^2 - x - a = 0$$

Watch out for (2)

• Invert before splitting
  → Continued fractions
• Drop the negative root
  → Infinite nested radicals