Indices and Surds: AFCAT Numerical Ability

Formulas (3)

• Product, quotient and power of a power · The three index laws
  $$a^m a^n = a^{m+n}, \quad \dfrac{a^m}{a^n} = a^{m-n}, \quad (a^m)^n = a^{mn}$$
• Zero and negative indices · Negative index
  $$a^{-n} = \dfrac{1}{a^n}, \qquad \left(\dfrac ab\right)^{-n} = \left(\dfrac ba\right)^{n}$$
• Simplifying fractions of letters with powers · Letter by letter
  $$\dfrac{p^a q^b}{p^c q^d} = p^{a-c}\, q^{b-d}$$

Watch out for (6)

• Add indices only on the same base
  → Product, quotient and power of a power
• A power of a power multiplies
  → Product, quotient and power of a power
• A negative index is not a negative number
  → Zero and negative indices
• A zero index gives 1, not 0
  → Zero and negative indices
• Minus a negative is plus
  → Simplifying fractions of letters with powers
• The coefficient takes the power too
  → Simplifying fractions of letters with powers

Formulas (4)

• A root is a fractional power · Root as an index
  $$\sqrt[n]{a} = a^{1/n}, \qquad \sqrt[m]{\sqrt[n]{a}} = a^{1/(mn)}$$
• Powers like m/n and decimal powers · Fractional index
  $$a^{m/n} = \left(\sqrt[n]{a}\right)^m = \sqrt[n]{a^m}$$
• Fractional indices on letters · Quotient with fractional indices
  $$\dfrac{x^{a/b}}{x^{c/d}} = x^{\frac ab - \frac cd}$$
• Equations in powers and in square roots · Equal powers of one base
  $$a^{x} = a^{k} \;\Rightarrow\; x = k \quad (a > 0,\ a \ne 1)$$

Watch out for (9)

• A half power is a square root, not half
  → A root is a fractional power
• Nested roots multiply their indices
  → A root is a fractional power
• A decimal index is a fraction, not a divisor
  → Powers like m/n and decimal powers
• Same index: multiply bases. Different indices: one base
  → Powers like m/n and decimal powers
• Root first keeps the numbers small
  → Powers like m/n and decimal powers
• The root of a cube is a three-halves power
  → Fractional indices on letters
• Subtract fractions over one denominator
  → Fractional indices on letters
• Isolate the root before squaring
  → Equations in powers and in square roots
• Match the bases before matching indices
  → Equations in powers and in square roots

Formulas (3)

• Simplest form: take out the largest square · Taking a square out
  $$\sqrt{k^2 a} = k\sqrt{a}$$
• Adding and subtracting like surds · Like surds
  $$m\sqrt a + n\sqrt a = (m + n)\sqrt a$$
• Multiplying and dividing surds · One root sign
  $$m\sqrt a \times n\sqrt b = mn\sqrt{ab}, \qquad \dfrac{\sqrt a}{\sqrt b} = \sqrt{\dfrac ab}$$

Reference tables (1)

Watch out for (8)

• The root of a perfect square is not a surd
  → What a surd is: order, pure and mixed, like and unlike
• The order is the index, not the number inside
  → What a surd is: order, pure and mixed, like and unlike
• Take out the LARGEST square
  → Simplest form: take out the largest square
• A decimal root halves the decimal places
  → Simplest form: take out the largest square
• The root does not add
  → Adding and subtracting like surds
• The square root of a sum is not the sum of the roots
  → Adding and subtracting like surds
• A root times itself is the number inside
  → Multiplying and dividing surds
• Finish in simplest form
  → Multiplying and dividing surds

Formulas (3)

• Squares and conjugate products of surds · Conjugate product
  $$(\sqrt p + \sqrt q)(\sqrt p - \sqrt q) = p - q$$
• Rationalising a denominator · Multiply by the conjugate
  $$\dfrac{1}{a + \sqrt b} = \dfrac{a - \sqrt b}{a^2 - b}$$
• The square root of a + 2√b · Square root of a surd expression
  $$\sqrt{a \pm 2\sqrt b} = \sqrt m \pm \sqrt n \quad (m + n = a,\ mn = b,\ m > n)$$

Watch out for (8)

• A square keeps its middle term
  → Squares and conjugate products of surds
• A conjugate product has no root left
  → Squares and conjugate products of surds
• Square the surd, not the number inside
  → Rationalising a denominator
• Multiply the top as well
  → Rationalising a denominator
• Change only the middle sign
  → Rationalising a denominator
• Make it 2√b before you look for the pair
  → The square root of a + 2√b
• With a minus sign, the larger root goes first
  → The square root of a + 2√b
• The root of a square is a modulus
  → The square root of a + 2√b