Zero and Negative Indices
Extend the index laws so that any non-zero number to the power 0 is 1 and a negative index means the reciprocal.
What a learner can do afterwards
- Rewrite an expression with negative indices in fraction form
- Explain why x⁰ = 1 by dividing two equal powers
- Combine the multiplication, division and power laws in one expression
1 · Read
Take x to the 5th divided by x to the 5th. Any number divided by itself is 1. The quotient rule turns the same division into x to the power 5 minus 5, which is x to the 0. Both ways describe one division, so x to the 0 must equal 1, for any nonzero number.
Now try x cubed divided by x to the 5th. The quotient rule gives x to the power 3 minus 5, which is x to the minus 2. Cancelling by hand leaves 1 over x squared. So a negative index means flip into a fraction and drop the minus sign: x to the minus 3 is 1 over x cubed.
Numbers work the same way. 5 to the minus 2 is 1 over 25, which is 0.04, and 2 to the minus 3 is 1 over 8, which is 0.125. With letters, multiply 2 x to the minus 4 by 3 x to the 6th: the numbers give 6 and the indices add to 2, so the answer is 6 x squared. When a power is raised to another power, you multiply the indices, so a to the minus 3 squared is a to the minus 6.
Keep the old laws running. Add indices when multiplying and subtract when dividing, then flip any negative leftover into a fraction. Try y to the minus 5 divided by y to the minus 2: subtract to get y to the minus 3, which is 1 over y cubed.
Zero power is 1, negative powers flip, and the old index laws keep working.
2 · Watch
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Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.