Rational (fractional) indices

Rational (fractional) indices

Rational Indices: Overview

  • Rational indices refer to indexes that are expressed as fractions or decimals.
  • They are also known as fractional indices.
  • Indices with rational exponents indicate that some form of root is being taken of a number.

Understanding Rational Indices

  • A number to the power of a fractional index, such as n^(1/2), represents the square root of n, while n^(1/3) denotes the cube root and so forth.
  • A number to a negative fractional power, like n^(-1/2), signifies the reciprocal of the square root of n.
  • In general, for any positive integer ‘a’ and ‘b’, n^(a/b) equals b-th root of n to the power ‘a’.

Working with Rational Indices

  • Multiplication rule: When multiplying numbers with same bases but different fractional indices, add the indices. For example, n^(a/b) * n^(c/d) = n^( (ad+bc)/bd).

  • Division rule: When dividing, subtract the indices in the same base. For example, n^(a/b) / n^(c/d) = n^( (ad-bc)/bd).

  • Power of power rule: When raising a power to another power, multiply the indices. For example, (n^(a/b))^c = n^(ac/b).

Calculating with Rational Indices

  • Calculations involving rational indices are often simplified by converting the fraction to its simplest form or by converting the index to a decimal.

Examples

  • In the expression 9^(1/2), can be simplified to 3, as 3 is the square root of 9.
  • For the expression 8^(2/3), it can be rewritten as (8^(1/3))^2. As the cube root of 8 is 2, this becomes 2^2 which is 4.

Conclusion

  • Rational or Fractional indices form an essential part of understanding exponents in algebra. They allow for the expression of roots and provide for simplified mathematical operations.