Negative indices

Negative Indices

Understanding the Concept

  • A number with a negative index represents the reciprocal of the number raised to that index when it’s positive.

  • This is a powerful concept in algebra and is used extensively in solving algebraic problems.

Working with Negative Indices

  • If a number ‘a’ has a negative index ‘n’, it is equivalent to 1 divided by (a^n), where ‘n’ is positive. In other words, a^-n equals 1/a^n.

  • For example, 2^-3 is equivalent to 1/2^3. Therefore 2^-3 equals 1/8.

Rules for Negative Indices

  • When multiplied, indices can be added together. For example, a^n * a^m equals a^(n+m).

  • Therefore, if a ∈ R and a ≠ 0, a^m * a^-n equals a^(m-n).

  • Similarly, a division law can be adapted for negative indices: a^m / a^n equals a^(m-n).

Applications of Negative Indices

  • Negative indices are extremely useful in dealing with expressions involving very large or very small numbers, like in scientific notation.

  • They also come into play when solving algebraic problems or simplifying algebraic expressions.

Examples

  • Consider the expression 5^-2. Following the rule of negative indices, this equals 1/5^2, hence simplifies to 1/25.

Remember

  • Avoid confusion between negative indices and negative bases. -3^2 is not the same as (-3)^2. The first expression equals -9, as the negative sign is not part of the base. The second expression equals 9, the square of -3.

Conclusion

  • Grasping the concept of negative indices is necessary for simplifying complex algebraic equations or expressions.

  • Check your work by using the definition of negative indices, converting back into a positive index to verify your solution.