Exponents and Negative Numbers
Purplemath
We've seen how to multiply negative numbers. Repeated multiplication is indicated by exponents, also called powers, orders, or indices.
To evaluate negative numbers raised to powers, we will use the cancellation-of-minus-signs property of multiplication.
What is the rule for squaring negative numbers?
When squaring a negative number, you are multiplying it by itself. In particular, you are multiplying two minusses. And the product of two minusses is plus, because the minus signs cancel off.
- Simplify (−3)2
The square means "multiplied against itself, with two copies of the base". This means that I'll have two minus signs, which I can cancel:
(−3)2
(−3)(−3)
(+3)(+3)
9
Pay careful attention and note the difference between the above exercise and the following:
- Simplify −32
The only difference between this expression and the one in the previous example is that, in this case, the minus sign is outside of the squaring; the power is only on the 3.
So, being outside of the squaring, the single solitary minus sign does not have anything against which to cancel. The minus will remain, and I expect my answer to be negative.
−32
−(3)(3)
−1(3)(3)
(−1)(9)
−9
In the second example, the square (that is, the "to the power 2" part) was only on the 3; it was not on the minus sign. Those parentheses in the first exercise make all the difference in the world!
Be careful with them, especially when you are entering expressions into software like an Excel spreadsheet or your graphing calculator. Different software may treat the same expression very differently, as one researcher has demonstrated quite thoroughly.
- Simplify (−3)3
For this simplification, the power is on the minus sign because of the parentheses.
Cubing (that is, raising to the power 3) means that there will be three minus signs. Two of them will cancel off, with the third minus sign remaining. So I expect my answer to be negative.
(−3)3 = (−3)(−3)(−3)
= (+3)(+3)(−3)
= (9)(−3)
= −27
- Simplify (−3)4
The parentheses tell me that the power is on the minus sign, along with the 3. The power is 4, so there will be four minus signs.
Minusses cancel off in pairs, so I expect all of the minus signs to cancel off, leaving me with a positive answer.
(−3)4 = (−3)(−3)(−3)(−3)
= (+3)(+3)(−3)(−3)
= (+3)(+3)(+3)(+3)
= (9)(9)
= 81
- Simplify (−3)5
The power is on the minus, because of the parentheses. The power means that there are five copies of −3 multiplied together. In particular, there are five minus signs.
Since minusses cancel off in pairs, I expect to have one minus sign left over, so my answer should be negative.
(−3)5 = (−3)(−3)(−3)(−3)(−3)
= (+3)(+3)(−3)(−3)(−3)
= (+3)(+3)(+3)(+3)(−3)
= (9)(9)(−3)
= −243
Note the pattern: A negative number taken to an even power gives a positive result (because the pairs of negatives cancel), and a negative number taken to an odd power gives a negative result (because, after cancelling, there will be one minus sign left over).
So if they give you an exercise containing something slightly ridiculous like (−1)1001, you know that the answer will either be +1 or −1, and, since 1001 is odd, then the answer will have to be −1.
You can also do negatives inside roots and radicals, but only if you're careful. You can simplify 
, because there is a number that squares to 16. That is,
...because 42 = 16. But what about 
? Can you square anything and have it come up negative? No! So you cannot take the square root (or the fourth root, or the sixth root, or the eighth root, or any other even root) of a negative number. On the other hand, you can do cube roots of negative numbers. For instance:
...because (−2)3 = −8. For the same reason, you can take any odd root (third root, fifth root, seventh root, etc.) of a negative number.
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