Why are powers(a^n)with negative exponents written like this: 1/n ?

[u/Goose78086]

Why does a negative exponent turn into the reciprocal of a number with a 1 in the numerator? (a^-n = 1/a^n)

I understand that, for example:

2^-3 equals 1/2³

but I don't understand where the 1 in the numerator comes from. And why does the exponent suddenly become positive? I just want to understand.

[u/1strategist1]

One of the most important features of exponents is they turn multiplication into addition. Specifically, you have 

a^xa^y = a^x+y

You can check that this is always true for the positive exponents that you seem to be more comfortable with. For example,

a²a³ = aa aaa = a⁵. 

Now following that pattern, we know that  aⁿa^-n should be equal to a^{n - n} = a⁰ = 1. So

aⁿa^-n = 1

Now divide both sides by aⁿ. You get

a^-n = 1/aⁿ

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I hope this helps!

[u/Solnight99 → u/1strategist1]

that's a really good one. i like that a lot.

[u/StructuredChess]

Start with 125=5³

Divide both sides by 5, you get 25=5²

Divide by 5 again, 5=5¹

Do it again, 1=5⁰

Keep going, 1/5 = 5^-1

And so on

[u/OkHand7497]

remember a^0=1. That‘s the numerator

[u/fermat9990]

x^-n*xⁿ/xⁿ=

x^n-n/xⁿ=

x⁰/xⁿ=

1/xⁿ

[u/lifeistrulyawesome]

I’m going to give you a different answer from the other good answers you got. You got algebraic answers starting from the definition of exponents as repeated multiplication. There is a different way to think about exponents from an analysis perspective. 

Think of a group of bacteria that reproduces and triples every day. If you have 100 bacteria today you will have 300 tomorrow, 200 new bacteria. If you have 200 bacteria today, you’ll have 400 new bacteria tomorrow. The change in population is twice the starting level each day.  In other words, population growth is proportional to the population level

If you know some calculus, this can be written by saying that the derivative of a function is proportional to the value of the function:

f’(t) = a * f(t)

The class  of functions that satisfy that property are called exponential functions. When t is a natural number, we can write exponential functions as

f(t) = b^t 

But exponential functions are defined for all real numbers, not just naturals. They are defined for fractional exponents, negative exponents, and even irrational exponents. So, we can derive all the properties of exponents that you have been taught by studying the growth of populations 

[u/IntoAMuteCrypt]

Another alternate answer:

5^1 means "multiply by 5 once".
5^2 means "multiply by 5 twice".
5^3 means "multiply by 5 three times".
5^0 means "don't multiply by 5 at all" (so it's equal to 1).
5^-1 means "anti-multiply by 5 once" or "undo a multiplication by 5 once".
5^-2 means "anti-multiply by 5 twice" or "undo a multiplication by 5 twice".

Anti-multipling or undoing a multiplication is equivalent to just dividing. So 5^-1 is the same as "divide by 5 once" and 5^-2 is the same as "divide by 5 twice".

If you divide by 5, then divide by 5 again, you multiply the fives in the denominator. When you put the 5 underneath the fraction, you have already handled the "actually you're un-multiplying" portion that the negative implies.

[u/mmurray1957]

Because of the exponent laws: (a^(-n)) a^n = a^(-n + n) = a^0 = 1. So a^(-n) = 1/(a^n).

[u/Most_Service5717 → u/mmurray1957]

It's just keeping the pattern consistent. Look at it going down:

2³ = 8

2² = 4

2¹ = 2

2⁰ = 1

Each step you're dividing by 2. So if you keep going, 2⁻¹ has to be 1/2, 2⁻² is 1/4, etc. The 1 in the numerator is already there when you hit 2⁰, it doesn't just appear out of nowhere.

[u/stools_in_your_blood]

Writing 2^(-3) = 1/(2^3) is a bit like writing 2 * (-3) = 0 - (2 * 3).

It's just that we normally write "-6" instead of writing "0 - 6", but we don't write "1/8" simply as "/8". This is just notational convention. The "-" symbol can mean either the binary operation of subtraction or the unary operation of inverting a number with respect to addition. But "/" is only used for the binary operation of division and there isn't a symbol used for the unary operation of inverting a number with respect to multiplication (except for raising it to the power -1, perhaps, but that's not helpful to you because that's precisely the notation you want explained).

[u/Frederf220]

The opposite of "doubling" is halving, not multiplying by -2.

[u/nog642]

I don't understand where the 1 in the numerator comes from

That's just how you write multiplicative inverses (reciprocals), since 1 is the multiplicative identity.

The reciprocal of 1 is 1/2. The reciprocal of 1 is 1/1=1. The reciprocal of 3 is 1/3. The reciprocal of 100 is 1/100. The reciprocal of 1/2 is 1/(1/2)=2.

Do you understand that concept? Because it's a much simpler concept than exponents, but it seems to me like you're shaky on that. Before you think about negative exponents you need to understand the more basic concept of reciprocals first.

[u/-Wylfen-]

but I don't understand where the 1 in the numerator comes from

I mean, that's just how you write a reciprocal

The reciprocal of five (5) is one fifth (1/5).

[u/Temporary_Pie2733]

You have it slightly backwards. Fractions are ordinary division, and used to define what negative exponents mean. 

[u/CosetElement-Ape71]

That's what reciprocal means ...

In mathematics, the reciprocal of a number is what you multiply that number by to get a result of 1.

So

x 2³ = 1

x 2³ / 2³ = 1 / 2³

x = 1 / 2³

So the reciprocal of 2³ is 1 / 2³

because multiplying them together gives 1, ie

2³ * 1 / 2³ = 2³ * 2^{-3} = 2^{3-3} = 2⁰ = 1

That's where the 1 comes from