Surds, Indices and Simplification: SSC CGL Quantitative Aptitude

Formulas (3)

• The order of operations (BODMAS) · Order of operations
  $$\text{Brackets} \to \text{Orders} \to \div,\ \times\ (\text{left to right}) \to +,\ -\ (\text{left to right})$$
• Removing nested brackets from the inside out · Minus before a bracket
  $$a - (b - c) = a - b + c \qquad a - (-b) = a + b$$
• The word 'of' and the bar in BODMAS · Of against times after a division
  $$a \div b \text{ of } c = \frac{a}{bc} \qquad a \div b \times c = \frac{ac}{b}$$

Watch out for (7)

• Multiplying before dividing
  → The order of operations (BODMAS)
• Adding before subtracting
  → The order of operations (BODMAS)
• Flipping only the first sign after a minus
  → Removing nested brackets from the inside out
• Losing a negative bracket
  → Removing nested brackets from the inside out
• Treating 'of' as an ordinary multiplication
  → The word 'of' and the bar in BODMAS
• Ignoring the bar
  → The word 'of' and the bar in BODMAS
• Doing 'of' before its own bracket
  → The word 'of' and the bar in BODMAS

Formulas (4)

• Fractions and mixed numbers in BODMAS · Fraction rules
  $$\frac{a}{b} \div \frac{c}{d} = \frac{ad}{bc} \qquad \frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd} \qquad p\frac{q}{r} = \frac{pr + q}{r}$$
• Working with decimals · Recurring decimals and handy divisions
  $$0.\overline{a} = \frac{a}{9} \qquad 0.\overline{ab} = \frac{ab}{99} \qquad x \div 0.25 = 4x \qquad x \div 0.5 = 2x$$
• Comparing and ordering fractions · Cross-multiplication test
  $$\frac{a}{b} > \frac{c}{d} \text{ exactly when } ad > bc \quad (b, d > 0)$$
• Continued fractions · Building one level
  $$a + \frac{1}{b} = \frac{ab + 1}{b} \qquad \frac{1}{\;p/q\;} = \frac{q}{p}$$

Watch out for (6)

• Multiplying mixed numbers part by part
  → Fractions and mixed numbers in BODMAS
• Turning the wrong fraction upside down
  → Fractions and mixed numbers in BODMAS
• Too few decimal places in a product
  → Working with decimals
• Expecting division to make a number smaller
  → Working with decimals
• Judging a fraction by the size of its numbers
  → Comparing and ordering fractions
• Starting a continued fraction from the top
  → Continued fractions

Formulas (1)

• Simplifying and evaluating expressions in letters · Powers of a negative number
  $$(-a)^{2} = a^{2} \qquad (-a)^{3} = -a^{3}$$

Watch out for (4)

• Pulling the unknown away from its multiplier
  → Finding a missing number in an expression
• Skipping the check
  → Finding a missing number in an expression
• Squaring a negative without brackets
  → Simplifying and evaluating expressions in letters
• Adding unlike terms
  → Simplifying and evaluating expressions in letters

Formulas (3)

• The laws of indices · Laws of indices
  $$a^m a^n = a^{m+n} \qquad \frac{a^m}{a^n} = a^{m-n} \qquad (a^m)^n = a^{mn} \qquad a^0 = 1 \qquad a^{-n} = \frac{1}{a^n}$$
• Square roots of perfect squares · Square of a number ending in 5
  $$(10n + 5)^2 = 100\,n(n + 1) + 25$$
• Nested square roots · Infinite nested roots
  $$\sqrt{n(n - 1) + \sqrt{n(n - 1) + \cdots}} = n \qquad \sqrt{n(n - 1) - \sqrt{n(n - 1) - \cdots}} = n - 1$$

Watch out for (7)

• Adding indices for different bases
  → The laws of indices
• Multiplying indices in a product
  → The laws of indices
• Reading a zero index as zero
  → The laws of indices
• Trusting the last digit alone
  → Square roots of perfect squares
• Taking the root of each term
  → Square roots of perfect squares
• Keeping the negative root
  → Nested square roots
• Using the shortcut on the wrong number
  → Nested square roots

Formulas (3)

• Difference of squares and cubes as a shortcut · Square and cube identities
  $$a^2 - b^2 = (a + b)(a - b) \qquad a^3 - b^3 = (a - b)(a^2 + ab + b^2) \qquad a^3 + b^3 = (a + b)(a^2 - ab + b^2) \qquad (a + b)^2 = a^2 + 2ab + b^2$$
• Sums of cubes in a fixed ratio · Sums of cubes in a fixed ratio
  $$\frac{a^3 + b^3}{(ka)^3 + (kb)^3} = \frac{1}{k^3}$$
• Common factors and quick answer checks · Common factor
  $$ab + ac = a(b + c) \qquad ab - ac = a(b - c)$$

Watch out for (6)

• Pairing a cube with the wrong middle sign
  → Difference of squares and cubes as a shortcut
• Dropping the middle term of a square
  → Difference of squares and cubes as a shortcut
• Using k instead of k cubed
  → Sums of cubes in a fixed ratio
• Pairing terms by position
  → Sums of cubes in a fixed ratio
• Treating a passed check as proof
  → Common factors and quick answer checks
• Losing the sign in a remainder check
  → Common factors and quick answer checks

Formulas (3)

• Simplifying, adding and expanding surds · Surd rules
  $$\sqrt{ab} = \sqrt{a}\,\sqrt{b} \qquad (\sqrt{a} + \sqrt{b})^2 = a + b + 2\sqrt{ab} \qquad (\sqrt{a} + \sqrt{b})^2 - (\sqrt{a} - \sqrt{b})^2 = 4\sqrt{ab}$$
• Rationalising with the conjugate · Conjugate
  $$(\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b}) = a - b \qquad \frac{1}{\sqrt{a} - \sqrt{b}} = \frac{\sqrt{a} + \sqrt{b}}{a - b}$$
• Comparing sums and differences of surds · Comparing surd sums and differences
  $$(\sqrt{a} + \sqrt{b})^2 = (a + b) + 2\sqrt{ab} \qquad \sqrt{a} - \sqrt{b} = \frac{a - b}{\sqrt{a} + \sqrt{b}}$$

Watch out for (6)

• Adding the numbers under the roots
  → Simplifying, adding and expanding surds
• Dropping the middle term of a surd square
  → Simplifying, adding and expanding surds
• Forgetting to divide by a minus b
  → Rationalising with the conjugate
• Missing a negative denominator
  → Rationalising with the conjugate
• Calling two surd sums equal because the numbers under the roots add up the same
  → Comparing sums and differences of surds
• Squaring when a side may be negative
  → Comparing sums and differences of surds

Formulas (3)

• The square root of a + 2√b · Square root of a surd
  $$\sqrt{a \pm 2\sqrt{b}} = \sqrt{m} \pm \sqrt{n} \quad \text{where } m + n = a,\ mn = b,\ m > n$$
• Equations with a sum and a difference of square roots · Sum times difference of roots
  $$(\sqrt{A} + \sqrt{B})(\sqrt{A} - \sqrt{B}) = A - B$$
• Componendo and dividendo · Componendo and dividendo
  $$\frac{p}{q} = \frac{a}{b} \;\Rightarrow\; \frac{p + q}{p - q} = \frac{a + b}{a - b} \qquad \frac{\sqrt{A} + \sqrt{B}}{\sqrt{A} - \sqrt{B}} = k \;\Rightarrow\; \frac{A}{B} = \left(\frac{k + 1}{k - 1}\right)^2$$

Watch out for (4)

• Not writing the surd part as 2√b first
  → The square root of a + 2√b
• Putting the smaller root first after a minus
  → The square root of a + 2√b
• Squaring a sum of roots term by term
  → Equations with a sum and a difference of square roots
• Forgetting to square the ratio of the roots
  → Componendo and dividendo