A question about Exponents

Question

I've been reading about Exponents, and I was wondering if there is a shorter way to do this same calculation, below:

2⁴ = 2 * 2 * 2 * 2 = 16

I keep seeing what seems to me a pattern in this, and other examples. The pattern I keep seeing (maybe it's a coincidence?), is that they always seem to only multiply the base number by the base number, until it reaches the same value that you would reach if you just multiplied the exponent by itself, just once.

What I mean is, can we not just do it this way instead? 2⁴ = 4 * 4 = 16

I.e. Instead of multiplying the base by the base a billion times, we just multiply the exponent by itself, once.

And how do we calculate the power of using just the Windows calculator?

Answer 1

The pattern is a only coincidence that occurs when the base is 2. Take an example when the base is 3:

$3^4 = 3*3*3*3 = 81$

$3^4$ is not $4*4 = 16$

Answer 2

What you saw is a bit of a coincidence since $2^2=4$.

What was used here is this: $$2^{4}=2^{2\cdot2}=(2^{2})^{2}=4^{2}=16$$

Answer 3

Your result is a coincidence. The reason it works for the example is that $2^4=(2^{2\cdot2})=(2^2)^2=4^2$. It is because $a^{bc}=(a^b)^c$. If speed in exponentiation is what you want, the standard divide and conquer algorithm will work. It goes like this: $a^{2b}=(a^b)^2$, $a^{2b+1}=a(a^b)^2$

For example

$2^{10}=(2^5)^2$

$2^5=2(2^2)^2$

$2^2=4$

$2^5=2.4.4=32$

$2^10=32.32=1024$

Note that this reduces the no. of multiplications. It may not be of much help to you as it is already implemented in all calculators, and is tedious to do by hand.

Answer 4

You certainly all know about the coincidence $$\frac{\not \!6 4}{1 \!\!\not \!6} = 4.$$