Squares, Cubes and Roots: AFCAT Numerical Ability

Formulas (2)

• Squaring a number, and the square identities · Square of a sum or a difference
  $$(a \pm b)^2 = a^2 \pm 2ab + b^2$$
• Testing a perfect square by prime factors · Perfect square test
  $$n = p^{a} q^{b} \cdots \text{ is a square when } a, b, \ldots \text{ are all even}$$

Reference tables (1)

Watch out for (6)

• The middle term
  → Squaring a number, and the square identities
• Brackets decide the sign
  → Squaring a number, and the square identities
• A good ending is not proof
  → The last digit of a square
• Squares ending in 5
  → The last digit of a square
• One odd power is enough to fail
  → Testing a perfect square by prime factors
• Factor all the way down
  → Testing a perfect square by prime factors

Formulas (4)

• Square root of a large perfect square · Root by bracketing
  $$(10a)^2 < N < (10a + 10)^2 \;\Rightarrow\; 10a < \sqrt N < 10a + 10$$
• Square roots in an expression · Roots of products and quotients
  $$\sqrt{ab} = \sqrt a\,\sqrt b, \quad \sqrt{\tfrac ab} = \dfrac{\sqrt a}{\sqrt b}$$
• Square roots of decimals and fractions · Root of a decimal
  $$\sqrt{\dfrac{a}{10^{2k}}} = \dfrac{\sqrt a}{10^{k}}$$
• Equations with a square root · Clearing a root
  $$\sqrt{ax + b} = k \;\Rightarrow\; ax + b = k^2$$

Watch out for (8)

• Two candidates for the units digit
  → Square root of a large perfect square
• Convert rupees to paise
  → Square root of a large perfect square
• No splitting over a sum
  → Square roots in an expression
• The root sign is positive
  → Square roots in an expression
• An odd number of decimal places
  → Square roots of decimals and fractions
• Halve the places, do not keep them
  → Square roots of decimals and fractions
• Square the whole side
  → Equations with a square root
• A root is never negative
  → Equations with a square root

Formulas (5)

• Cubing a number, and the cube identities · Sum and difference of cubes
  $$a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)$$
• Cube roots by prime factors · Cube root by prime factors
  $$\sqrt[3]{p^{3a} q^{3b}} = p^{a} q^{b}$$
• Cube roots of negative numbers, and cube root equations · Clearing a cube root
  $$\sqrt[3]{ax + b} = k \;\Rightarrow\; ax + b = k^3$$
• Cube roots of decimals and fractions · Cube root of a decimal
  $$\sqrt[3]{\dfrac{a}{10^{3k}}} = \dfrac{\sqrt[3]{a}}{10^{k}}$$
• Square roots and cube roots in one expression · One root at a time
  $$\sqrt{a^2} = a, \quad \sqrt[3]{b^3} = b$$

Reference tables (1)

Watch out for (10)

• An odd power keeps the sign
  → Cubing a number, and the cube identities
• A cube of a sum is not a sum of cubes
  → Cubing a number, and the cube identities
• Cubes can end in any digit
  → The cubes from 1 to 10, and spotting a perfect cube
• Bracket with cubes, not squares
  → The cubes from 1 to 10, and spotting a perfect cube
• Groups of three, not pairs
  → Cube roots by prime factors
• Cube, do not square
  → Cube roots of negative numbers, and cube root equations
• A negative cube root exists
  → Cube roots of negative numbers, and cube root equations
• Decimal places in threes
  → Cube roots of decimals and fractions
• A third of the places
  → Cube roots of decimals and fractions
• Read the small index
  → Square roots and cube roots in one expression

Formulas (2)

• Least multiplier or divisor, by prime factors · What a perfect power needs
  $$\text{square: every exponent even}, \quad \text{cube: every exponent a multiple of } 3$$
• Least number to add or subtract for a perfect square · Nearest squares
  $$k^2 \le n < (k + 1)^2: \quad \text{add } (k + 1)^2 - n, \quad \text{subtract } n - k^2$$

Watch out for (5)

• Least means only the missing primes
  → Least multiplier or divisor, by prime factors
• A cube needs threes
  → Least multiplier or divisor, by prime factors
• Read the verb
  → Least multiplier or divisor, by prime factors
• Which neighbour
  → Least number to add or subtract for a perfect square
• Step from a round square
  → Least number to add or subtract for a perfect square

Formulas (4)

• Multiplying and dividing roots · Product and quotient rules
  $$\sqrt[n]{a}\,\sqrt[n]{b} = \sqrt[n]{ab}, \quad \dfrac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\dfrac ab}$$
• Simplest form of a surd, and like surds · Taking out a square or cube factor
  $$\sqrt{a^2 b} = a\sqrt b, \quad \sqrt[3]{a^3 b} = a\sqrt[3]{b}$$
• Squaring a two-term surd, and its square root · Square of a two-term surd
  $$(\sqrt a \pm \sqrt b)^2 = a + b \pm 2\sqrt{ab}$$
• Rationalising a denominator · Rationalising with the conjugate
  $$\dfrac{1}{\sqrt a + \sqrt b} = \dfrac{\sqrt a - \sqrt b}{a - b}$$

Watch out for (8)

• Products combine, sums do not
  → Multiplying and dividing roots
• One kind of root at a time
  → Multiplying and dividing roots
• Take out the largest square
  → Simplest form of a surd, and like surds
• Unlike surds stay apart
  → Simplest form of a surd, and like surds
• Keep the cross term
  → Squaring a two-term surd, and its square root
• Larger first with a minus
  → Squaring a two-term surd, and its square root
• Multiply the top too
  → Rationalising a denominator
• A cube root needs two more
  → Rationalising a denominator

Formulas (3)

• A root as a fractional power · Fractional power
  $$a^{m/n} = \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m$$
• The laws of indices · Laws of indices
  $$a^m a^n = a^{m + n}, \quad \dfrac{a^m}{a^n} = a^{m - n}, \quad (a^m)^n = a^{mn}$$
• Equations with equal powers · Equal powers
  $$a^x = b^y = k \;\Rightarrow\; a = k^{1/x},\ b = k^{1/y}$$

Watch out for (6)

• The bottom is the root
  → A root as a fractional power
• A half power is not half
  → A root as a fractional power
• Add, do not multiply
  → The laws of indices
• A negative index is not a negative number
  → The laws of indices
• Same base first
  → Equations with equal powers
• A product becomes a sum
  → Equations with equal powers