Using indices, we can show a number times itself many times or as another way of writing a square or cube root. Indices make complex calculations that involve powers easier.
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 reciprocalThe reciprocal of a number is 1 divided by that number. The reciprocal of a fraction is that fraction turned upside down, eg the reciprocal of 3/4 is 4/3. (\(a^{-m} = \frac{1}{a^m}\) 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).