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.
More to read
Zero and Negative Exponent Rules
When you multiply powers with the same base, you add the exponents. That pattern is what makes zero and negative exponents make sense.
Take x^5 divided by x^5. Any number divided by itself is 1, so the answer is 1. But the quotient rule says x^5 divided by x^5 equals x^(5-5), which is x^0. Since both ways give the same division, x^0 must equal 1. This works for any nonzero number, not just x.
Now try x^3 divided by x^5. Using the quotient rule, that is x^(3-5), or x^-2. You can also cancel three of the x's on top with three on the bottom, leaving 1 over x times x, which is 1/x^2. So x^-2 and 1/x^2 are the same thing. A negative exponent means 'flip the base into a fraction and drop the minus sign.'
The two rules: - Any nonzero number to the power 0 is 1. - A negative exponent means take the reciprocal and make the exponent positive.
Raising to the Zero and Negative Powers
The quotient rule says that when you divide two powers with the same base, you subtract the exponents: c^m divided by c^n equals c^(m-n). That works fine when m is bigger than n. But what happens when m equals n, or when n is bigger than m?
If m equals n, like c^4 divided by c^4, the division just gives 1, because anything divided by itself is 1. The quotient rule says this should equal c^(4-4), which is c^0. For both answers to match, c^0 has to be 1.
If n is bigger than m, like c^2 divided by c^5, the quotient rule gives c^(2-5), or c^-3. Canceling factors of c by hand shows this equals 1/c^3. So a negative exponent means the reciprocal of the positive power.
Defining c^0 = 1 and c^-n = 1/c^n this way keeps the product and quotient rules working for every whole number exponent, not just the positive ones.
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