Proof that (-3)0 = 1
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You can present the same pattern for other numbers, too. Once your child discovers the rule for this sequence is division by -3 at each step, then the next logical step is that (-3)0 = 1.
The other proof idea is to first notice the following rule about multiplication (n is any integer):
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Mathematics is logical and the mathematical rules work in (nearly) all cases. You have noticed how many times a rule is stated to apply 'for any integer n' or for 'all whole numbers'. That makes mathematics beautiful. So suppose we don't know what (-3)0 is. Whatever (-3)0 is, if it obeys this rule above, then
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(-3)7 × (-3)0 = (-3)7 + 0
In other words, (-3)7 × (-3)0 = (-3)7 |
(-3)3 × (-3)0 = (-3)3 + 0
In other words, (-3)3 × (-3)0 = (-3)3 |
(-3)15 × (-3)0 = (-3)15 +
0
In other words, (-3)15 × (-3)0 = (-3)15 |
...and so on for all kinds of possible exponents.
( In fact, we could write that (-3)x × (-3)0 = (-3)x, where x is any whole number. That is the way you see it written in some math books, because whenever something is true for all kinds of numbers, it is written using a letter to symbolize those 'all kinds of numbers' .)
Since we are supposing that we don't yet know what (-3)0 is, let's substitute P for it. Now look at the equations we found above. Knowing what you know about properties of multiplication, what kind of number can P be?
| (-3)7 × P = (-3)7 | (-3)3 × P = (-3)3 | (-3)15 × P = (-3)15 |
So... What is the only number that when you multiply by it, nothing changes?
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Question. What is the difference between -1 to the zero power and (-1) to the zero power? Will the answer = 1 for both? Example 1: -10 = ____ Example 2: (-1)0 = ___ Answer: As already explained, the answer to (-1)0 is 1 since we are raising the number Another example: in the expression -(-3)2 , the first negative sign means you need to take the opposite number of whatever the rest of the expression comes to. So since (-3)2 = 9, then -(-3)2 = -9. |
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Question. Why is zero with a zero expoent come up with an error?? Please explain why it doesn't exist. In other words, what is 00? Answer:Zero to zeroth power is often said to be "an indeterminate form", because it could have several different values.
See also these articles from Dr. Math: n^0 Power = 1: Defined or Proved? |
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What is the difference between power and the exponent?
Varthan Power is the whole thing, or the answer. Exponent is the little number. For example, 8 is a power (of 2) since 23 = 8. 3 is the exponent. See also Powers and exponents at FactMonster.com. |
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