# What is the easiest way to compare two powers with different bases, signs, and exponants?

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### 1 Answer

An example would help clarify this question.

How I'm reading it, you want to know a quick way to tell which quantity is larger when given something like `12^10` vs. `10^12`

Let's look at some smaller examples.

`2^3<3^2`

Maybe the larger base always wins?

`2^5>5^2`

I guess not.

Exponents are crazy powerful. An exponential function will always pass up a linear function, though sometimes it's hard to guess when. Look at this: y = 10x vs. y = 1.1^x. The red graph is very steep, and it's hard to believe that the orange graph will ever catch up and pass it.

But if we zoom far out:

Now I know you're not asking about linear quantities, but this just shows how powerful the exponent is.

Let's look back at `10^12` vs `12^10`

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You know they're each larger than `10^10` , but can we get an idea of how much larger?

`10^12=10^10*10^2` so this one's 100 times larger.

`12^10=(10*1.2)^10=10^10*1.2^10` Since 1.2 is pretty close to 1, raising it to the 10th power won't make it that big.

So 10^12 wins.

There aren't any good rules that I know of for comparing exponential quantities, but remember that a bigger exponent can have a dramatic effect.

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