An index, or power, is the small floating number that appears after a number or letter. Indices show how many times a number or letter has been multiplied by itself.
simplifyA fraction is simplified when there are no more common factors shared by the numerator and denominator. For example, the fraction 8/10 simplifies to 4/5 by dividing the numerator and denominator by the common factor of 2.\(d^4 \div d^5\).
Dividing indices means subtracting the powers.
\(d^4 \div d^5 = d^{4 - 5} = d^{-1}\). This is an example of a negative index.
But \(d^4 \div d^5\) also equals \(\frac{d \times d \times d \times d}{d \times d \times d \times d \times d}\). Cancelling common factorA whole number that divides into two (or more) other numbers exactly, eg 4 is a common factor of 8, 12 and 20. gives \(\frac{\cancel{d} \times \cancel{d} \times \cancel{d} \times \cancel{d}}{\cancel{d} \times \cancel{d} \times \cancel{d} \times \cancel{d} \times d}\), which gives \(d^4 \div d^5 = \frac{1}{d}\).
So \(d^{- 1} = \frac{1}{d}\).
The rule for negative indices is \(a^{-m} = \frac{1}{a^m}\)
A negative power is often referred to as a reciprocal (\(a^{-m} = \frac{1}{a^m}\) is the reciprocal of \(a^m\)).
Examples
\(p^{-2} = \frac{1}{p^2}\)
\(2^{-3} = \frac{1}{2^3} = \frac{1}{8}\)
\(10^{-2} = \frac{1}{10^2} = \frac{1}{100}\) (you may recognise this notation from standard form)