Negative Indices

Look at the powers of two counting down: two cubed is eight, two squared is four, two to the one is two, and two to the zero is one. Each time the index drops by one, the value halves. Keep the pattern going. Half of one is a half, so two to the minus one is one half. Half again is a quarter, so two to the minus two is one quarter, which is one over two squared. A negative index does not make the number negative. It means one over the positive power.

In general, a to the minus n means one over a to the n. Ten to the minus three is one over ten cubed, which is one thousandth, nought point nought nought one. Four to the minus two is one over sixteen. For a fraction, the reciprocal is the fraction turned upside down, so two fifths to the minus two is five halves squared, which is twenty-five quarters.

The index laws work for negative indices exactly as before; you just need care with the signs. Three to the five times three to the minus seven: add the indices. Five plus minus seven is minus two, so the answer is three to the minus two, which is one ninth. Two to the minus three divided by two to the minus five: subtract. Minus three take away minus five is minus three plus five, which is two, so the answer is two squared, which is four. For a power of a power, multiply: five to the minus two, all to the minus one, is five squared, twenty-five.

With letters, a final answer is written with positive indices. x to the minus three is one over x cubed. In three a to the minus two, the index sits only on the a, so the a moves to the bottom and the three stays on top: three over a squared. In three a in brackets to the minus two, the index sits on the whole bracket, so the whole bracket moves down and is squared: one over nine a squared. And a base with a negative index on the bottom moves to the top: one over a to the minus one is just a.

To simplify an expression with negative indices, use the index laws first and write positive indices only at the end. Six a squared b to the minus three, times two a to the minus five b: six times two is twelve; for a, two plus minus five is minus three; for b, minus three plus one is minus two. That gives twelve a to the minus three b to the minus two. Now move the letters with negative indices below the line: twelve over a cubed b squared. The twelve stays on top. Quiz and exercises now.