Exponents - Why is 2 to the power of 1 not 4?

[MathHelp]

Hi team,

I'm really struggling with understanding how exponents make sense. According to a basic google definition:

"An exponent refers to the number of times a number is multiplied by itself."

This would indicate to me that 2 to the power of one would mean that 2 needs to be multiplied by itself... once. So 2 x 2 is 4. But I know that 2 to the power of 1 is actually one.

Why is 2 squared not 2x2x2? I can only see two "x"s in there...

Thanks for your advice.

[Markus Hanke]

8 hours ago, MathHelp said:

An exponent refers to the number of times a number is multiplied by itself.

That’s not a very good definition, IMHO - especially since the exponent can be any number, even a negative one, a fraction, an irrational one, or a complex number.

Let’s stick to simple, natural numbers like 1,2,3,... for now. Exponentiation is then a short-hand notation for a multiplicative series starting at 1, followed by as many multiplications with the base number as indicated in the exponent:

1 x ... x ... x ... and so on

Thus:

2^0 means you start at 1, followed by no further multiplications. Thus 2^0=1.

2^1 means you start at 1, followed by exactly one multiplication by 2. Thus 2^1=1x2=2.

2^2 means you start at 1, followed by two consecutive multiplications by 2. Thus 2^2=1x2x2=4.

2^3 means you start at 1, followed by three consecutive multiplications by 2. Thus 2^2=1x2x2x2=8.

And so on.

Does this make sense now?

[MathHelp]

That makes perfect sense. Thank you so much!

[studiot]

12 hours ago, MathHelp said:

"An exponent refers to the number of times a number is multiplied by itself."

 

+1 to Markus for pointing out that the full definition of exponents needs to work for any number.

and I agree that the definition you have found is absolutely crap.

However it is more usual to introduce exponents by not saying the number of multiplications performed but the number of '2s' multiplied together.

You don't then need to go into that long series involving 1.

The exponenent n is the number of original number multiplied together.

For example

One two on its own   =  2  =  2¹ = 2

Two twos  multiplied together  =  2 x 2  = 2²  = 4    note this is only one multiplication

Three twos multiplied together  = 2 x 2 x 2 = 2³  = 8  note this is only two multiplications

and so on.

So the number of multiplications is always one less than the numer of 2s

 

You can then go on to find out what 2⁰  and 2^-1 mean and finally fractional and decimal exponents

 

[MathHelp]

This is all good stuff, it is going to make understanding a lot of other maths ideas a whole lot easier.

I always new 2² was 4 and how to calculate exponents, I just never understood how it made sense (I have come across many bad definitions). Because the definitions never made sense, it also meant things like logarithms and square roots never made sense to me either.