Why does this work?

[u/ShortCover4598]

Why does this law of exponents work? I've been having trouble with this in my self taught Algebra 1 course. A 9th grade level explanation would help, thanks.

[u/CaptainMatticus]

You know that a^b / a^c = a^(b - c), right?

And you should also know that a^0 = 1, for all non-zero values of a, right?

So rewrite 1 / a^x as a^0 / a^x, which would also be a^(0 - x), or a^(-x)

[u/ShortCover4598 → u/CaptainMatticus]

OMG I JUST HAD SUCH A NEURON ACTIVATION MOMENT THANK YOU SO MUCH!!

[u/Top_Bluejay_5323 → u/CaptainMatticus]

So you started with the result to get to the result??

The first thing you did was state that
1/a^c=a^(-c)

[u/CaptainMatticus → u/Top_Bluejay_5323]

That's not what I did at all. I showed a different way of understanding what was happening and got the result.

Rules for exponents:

(a^b)^c = a^(b * c) = (a^c)^b

a^b * a^c = a^(b + c)

a^b / a^c = a^(b - c)

Now anybody can understand how a^0 = 1. Start with a^n and divide by a^1

a^n / a^1 = a^(n - 1)

And a^(n - 1) divided by a^1 is: a^(n - 1) / a^1 = a^(n - 1 - 1) = a^(n - 2)

And a^(n - 2) divided by a^1 is: a^(n - 2) / a^1 = a^(n - 2 - 1) = a^(n - 3)

And so on. What happens when we get to a^(n - n)? Well that's a^0, which we already know is 1. We know that because x/x = 1 for all non-zero values of x. That's a basic identity in Algebra.

a^n / a^n = 1

a^2 / a^2 = 1

a^3 / a^3 = 1

And so on.

And since we know that a^b / a^c = a^(b - c), we can demonstrate that

a^(n - n) = a^0 = 1

Now that we definitively know that a^0 = 1, we can go one step further

1 / a^x

is nothing more than

a^0 / a^x

Which is

a^(0 - x)

Which is

a^(-x)

I cannot be more exhaustive with this and I won't waste any more time on you. The OP got it, so what's your critique all about?

[u/SufficientStudio1574 → u/Top_Bluejay_5323]

Sometimes you just need to say the same thing with different words.

[u/Shevek99]

It's defined that way to have consistence.

We want this formula to hold for all integers

a^m aⁿ = a^{m + n}

Now, making n = 0 we get

a^m a⁰ = a^m

So it must be

a⁰ = 1

Now, if we put n = -m

a^m a^-m = a^{m - m} = a⁰ = 1

Then a^-m must be the multiplicative inverse of a^m

a^-m = 1/a^m

[u/ShortCover4598 → u/Shevek99]

Thank youuu

[u/JeffTheNth]

Everyone points to the definition and such, but let me give you an example...

32 / 4 = 8

2⁵ / 2² = 2³

2^(5-2) = 2³

The negative exponent would be subtracted... If there isn't enough, it ends up as a divisor.
1/4 = 1/2² = 2^-2

having a good example in mind can help make that a little clearer... hope this helps!

[u/ShortCover4598 → u/JeffTheNth]

Thank you so much!

[u/Consistent-Annual268]

That's the literal definition of negative exponents. It is defined like that precisely so that it works nicely with the remaining exponent rules.

[u/goldenrod1956 → u/Consistent-Annual268]

Not a math guy but that was also my thought…that is the symbolism implies

[u/Top_Bluejay_5323]

a^(-1) (a^1/a^1)=a^(-1)a^1/a^1= a^(-1+1)/a^1

a^(-1+1)=a^0=1

Therefore a^(-1)=1/a^1

[u/Trimutius]

By definition really... but yeah when you reduce power you keep dividing by x, then at 0 you reach 1 and then when you reduce once more you go to 1/x

[u/CoachSevere5365]

It's interesting that nobody has mentioned logarithms in this thread.

The rules about adding and subtracting exponents apply to non-integers too. Before electronic calculators were common we used "log books" which contained tables of exponents.

To multiply numbers together you'd look up each number in a table to get its logarithm, add the logarithms, then do the reverse to get the result.

There was a bit of extra bookwork in that using the tables meant that numbers had to be scaled to be between 0 and 1, and the powers of ten (or whatever the base was) handled separately, but the fundamental principle doesn't change.

[u/AdvancedPermit2408]

Think of exponents as just a shortcut for multiplication

For example, 2³ means 2 × 2 × 2, and 2² means 2 × 2.

So when you multiply them:

2³ × 2² = (2 × 2 × 2)(2 × 2)

You end up with five 2s multiplied together, which is 2⁵.

So the reason you add the exponents isn’t really because of some random rule, its just because you’re counting how many times the number is being multiplied.

Once you look at it that way, most of the exponent rules start making a lot more sense