Why something 'power' zero is one like a⁰ = 1?
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mmm4444bot
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The simple answer is because that definition works with all of the other properties of exponents.
Here's another way to think about it: Every Real number has a "hidden" factor of 1 in front of it:
4 is 1∙4\(\displaystyle \quad\quad\)⅞ is 1∙⅞\(\displaystyle \quad\quad\)∜3 is 1∙∜3\(\displaystyle \quad\quad\)θ is 1∙θ\(\displaystyle \quad\quad\)∑xᵢ is 1∙∑xᵢ\(\displaystyle \quad\quad\)ℝ is 1∙ℝ
When I say "hidden", I mean unwritten. There are many instances where we have factors of 1, but we don't write them (see post #2, if you're interested).
Powers are no different:
1∙332 = 1∙33∙33\(\displaystyle \quad\quad\)1∙x5 = 1∙x∙x∙x∙x∙x
See how the exponent tells us the number of base factors, in the power?
1∙a0 = ?
When there are zero base factors, all that's left is the unwritten factor of 1 -- which is no longer hidden!
1∙a0 = 1
a0 = 1
Well, it's no longer hidden on the right-hand side, that is.
Here's another way to think about it: Every Real number has a "hidden" factor of 1 in front of it:
4 is 1∙4\(\displaystyle \quad\quad\)⅞ is 1∙⅞\(\displaystyle \quad\quad\)∜3 is 1∙∜3\(\displaystyle \quad\quad\)θ is 1∙θ\(\displaystyle \quad\quad\)∑xᵢ is 1∙∑xᵢ\(\displaystyle \quad\quad\)ℝ is 1∙ℝ
When I say "hidden", I mean unwritten. There are many instances where we have factors of 1, but we don't write them (see post #2, if you're interested).
Powers are no different:
1∙332 = 1∙33∙33\(\displaystyle \quad\quad\)1∙x5 = 1∙x∙x∙x∙x∙x
See how the exponent tells us the number of base factors, in the power?
1∙a0 = ?
When there are zero base factors, all that's left is the unwritten factor of 1 -- which is no longer hidden!
1∙a0 = 1
a0 = 1
Well, it's no longer hidden on the right-hand side, that is.
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mmm4444bot
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Two mor examples where "hidden" factors of 1 are written is factoring and simply expressing x (or whatever name you prefer). :cool:
2∙x + 2 = 2∙(x + 1)
Multiply a by an : a ∙ an = a1+n
Parents: Here are two presentations for students (one pre-algebra; another beginning algebra).
https://www.youtube.com/watch?time_continue=2&v=b9q24AS2mR0
2∙x + 2 = 2∙(x + 1)
Multiply a by an : a ∙ an = a1+n
Parents: Here are two presentations for students (one pre-algebra; another beginning algebra).
https://www.youtube.com/watch?time_continue=2&v=b9q24AS2mR0
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Dr.Peterson
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A way to make it seem particularly reasonable is to think of any whole-number power xn as "start with 1 and multiply by x, n times":
x3 = 1*x*x*x
x2 = 1*x*x
x1 = 1*x
x0 = 1
This makes better sense than the common "multiply x by itself n times", since this way there really are n multiplications.
HallsofIvy
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Another way of looking at it:
For n a positive integer, \(\displaystyle x^n\) is defined as "x multiplied by itself n times". That is, \(\displaystyle x^3= x\cdot x\cdot x\). From that we can see that \(\displaystyle x^{n+ m}\), "x multiplied by itself m+ n times", is the same as "x multiplied by itself m times" and then "x multiplied by itself n times": \(\displaystyle (x^n)(x^m)\). So we have \(\displaystyle (x^n)(x^m)= x^{n+ m}\), a very useful formula. We would like that formula to work for other, non-positive integer, powers of x, in particular \(\displaystyle x^0\). "0" has the property that it is the "additive identity"- for any n, n+ 0= n. So we want to have \(\displaystyle x^{n+ 0}= x^n= (x^n)(x^0)\). That will be true if we define \(\displaystyle x^0= 1\).
For n a positive integer, \(\displaystyle x^n\) is defined as "x multiplied by itself n times". That is, \(\displaystyle x^3= x\cdot x\cdot x\). From that we can see that \(\displaystyle x^{n+ m}\), "x multiplied by itself m+ n times", is the same as "x multiplied by itself m times" and then "x multiplied by itself n times": \(\displaystyle (x^n)(x^m)\). So we have \(\displaystyle (x^n)(x^m)= x^{n+ m}\), a very useful formula. We would like that formula to work for other, non-positive integer, powers of x, in particular \(\displaystyle x^0\). "0" has the property that it is the "additive identity"- for any n, n+ 0= n. So we want to have \(\displaystyle x^{n+ 0}= x^n= (x^n)(x^0)\). That will be true if we define \(\displaystyle x^0= 1\).
mmm4444bot
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Thank you!!
Everytime I see or hear somebody say something like, "x^2 is x multiplied by itself two times", something inside me cringes. heh
x^2 = x ͏∙ x
The only way I can describe the right-hand side as two multiplications (and still be happy) is to recall that factor of 1 in front.
x^2 = 1 ∙ x ͏∙ x
mmm4444bot
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That's a good example; it shows one instance of how the definition (30=1) fits nicely with other properties of exponents because:
32 ͏∙ 3-2 = 32-2
:cool: