Surds, Indices and Simplification: SPI Mathematics

Formulas (2)

• Multiplying, dividing and raising powers · Laws of indices
  $$a^m \times a^n = a^{m+n} \qquad a^m \div a^n = a^{m-n} \qquad (a^m)^n = a^{mn} \qquad (ab)^n = a^n b^n$$
• Zero and negative indices · Zero and negative indices
  $$a^0 = 1 \ (a \ne 0) \qquad a^{-n} = \frac{1}{a^n} \qquad \left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^{n}$$

Watch out for (6)

• Adding indices for a power of a power
  → Multiplying, dividing and raising powers
• Joining different bases
  → Multiplying, dividing and raising powers
• Spreading a power over a sum
  → Multiplying, dividing and raising powers
• Reading a zero index as 0
  → Zero and negative indices
• Reading a negative index as a negative number
  → Zero and negative indices
• Losing the sign of a negative base
  → Zero and negative indices

Formulas (2)

• Fractional indices as roots · Fractional index
  $$a^{\frac{1}{n}} = \sqrt[n]{a} \qquad a^{\frac{p}{q}} = \left(\sqrt[q]{a}\right)^{p}$$
• Solving equations in powers · Equal powers of one base
  $$a^x = a^y \Rightarrow x = y \quad (a > 0,\ a \ne 1)$$

Watch out for (6)

• Swapping the root and the power
  → Fractional indices as roots
• Forgetting to turn a fraction over
  → Fractional indices as roots
• Miscounting the decimal places
  → Fractional indices as roots
• Matching indices on different bases
  → Solving equations in powers
• Treating a difference of powers as one power
  → Solving equations in powers
• Missing the answer x = 1
  → Solving equations in powers

Formulas (3)

• Simplifying and comparing roots · Roots as powers
  $$\sqrt{ab} = \sqrt{a}\,\sqrt{b} \qquad \sqrt[n]{a} = a^{\frac{1}{n}}$$
• Rationalising the denominator · Conjugate pair
  $$(a + \sqrt{b})(a - \sqrt{b}) = a^2 - b \qquad (\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b}) = a - b$$
• Square root of a surd and endless roots · Square root of a surd and an endless root
  $$\sqrt{(m + n) + 2\sqrt{mn}} = \sqrt{m} + \sqrt{n} \qquad x = \sqrt{k + x} \Rightarrow x^2 - x - k = 0$$

Watch out for (9)

• Splitting a root over a sum
  → Simplifying and comparing roots
• Getting the decimal places wrong
  → Simplifying and comparing roots
• Adding unlike surds
  → Simplifying and comparing roots
• Multiplying only the denominator
  → Rationalising the denominator
• Losing a negative denominator
  → Rationalising the denominator
• Using the wrong square identity
  → Rationalising the denominator
• Not making the root term 2 root b first
  → Square root of a surd and endless roots
• Keeping the negative root
  → Square root of a surd and endless roots
• Putting the smaller number first
  → Square root of a surd and endless roots