Indices

Understanding Indices

  • Indices (also referred to as powers or exponents) are used to express numbers that have been multiplied by themselves a certain number of times.
  • A number, say a, raised to the power of n (expressed as a^n) means n is the index and a is the base. a^n represents a multiplied by itself n times.

Basic Rules of Indices

  • Product rule of indices: For any positive number a, a^m * a^n = a^(m + n). This rule states that when multiplying two powers of the same base, you can add the exponents.
  • Quotient rule of indices: For any nonzero number a, a^m / a^n = a^(m - n). This rule states that when dividing two powers of the same base, subtract the exponents.
  • Power of power rule: For any positive number a, (a^m)^n = a^(mn). This rule states when raising a power of a number to another power, multiply the exponents.
  • Zero exponent rule: Any nonzero number raised to the power of zero is always 1.

Indices and Roots

  • The nth root of a number a is the inverse operation of raising a to the power n.
  • The square root (√) of a number a is equal to a^0.5. The cube root (∛a) of a number a is equal to a^(1/3).
  • The Negative index rule: For any nonzero number a, a^-n = 1/a^n. This rule states that the negative index of a number equals the reciprocal of that number raised to the corresponding positive index.

Key Concepts in Indices

  • Rational indices: A number raised to a fraction m/n (expressed as a^(m/n)) is equivalent to the nth root of a raised to the mth power.
  • Surd form: Rather than express a number in fractional indices, it can be written in surd form. For instance a^(1/2) is √a and a^(1/3) is ∛a.

Common Pitfalls in Indices

  • Remember that multiplying indices applies only when the base is the same. Failing to understand this is a common mistake.
  • Note that a^0 is always 1 for any nonzero number a, don’t assume that any number raised to power zero is zero.
  • Be sure to differentiate between the reciprocal of a base (from negative indices) and the reciprocal of a power. For instance, (2^-3) is 1/8 not -1/8.
  • Always remember to practise problems related to indices. Understanding the concepts and rules of indices are fundamental to solving more complex numerical and algebraic problems.