Why something 'power' zero is one like a⁰ = 1?

[Indranil]

Why something 'power' zero is one like a⁰ = 1?

 

[mmm4444bot]

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.

 

[mmm4444bot]

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

 

[Dr.Peterson]

Why something 'power' zero is one like a⁰ = 1?

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]

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 $\displaystyle x^0= 1$.

 

[mmm4444bot]

x^n …

start with 1 and multiply by x, n times …

… This makes better sense than the common "multiply x by itself n times", since this way there really are n multiplications.

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

 

[sinx]

Why something 'power' zero is one like a⁰ = 1?

because 3²=9, and 3^-2=1/9
3²*3^-2=9/9=3⁰​=1.

 

[mmm4444bot]

… 3² = 9 and 3^-2 = 1/9

3² * 3^-2 = 9/9 = 3⁰ ​= 1

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 = 3^2-2

:cool: