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Evaluate the expression: 4n to the first power minus 2n to the zeroth power, for n equals 1 and n equals 5.

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So let's do n equals 1 first.
So every place we see an n let's put a one in there.

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So this is equal to 4 times 1 to the first power minus 2 times 1 to the zero power.

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Every place we saw an n, we have an n there. We have an n to the first. n to the first, 1 to the first.

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So remember, we want to do our order of operations. We have PEMDAS, and I'm going to write it this way just to make it clear that we do

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parentheses, then exponentiation, then multiplication and division are at the same level. They get the same priority.

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And then addition and subtraction get the same priority.

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So we want to do our exponents before we do anything else. There are no parentheses here.

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So we first want to evaluate 1 to the 1st power, and anything to the first power is just that anything.

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If I just have any number to the first power, that is just going to be equal to that number.

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So 1 to the first power is just equal to 1.

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So this is going to be 4 times 1. Let me write it:
4 times 1, minus 2 times 1 to the zeroth power.

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Now any number other than zero to the zeroth power is going to be 1.

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So if I tell you x to the zeroth power is equal to what, and I tell you that x is not equal to zero, what is x to the zeroth power?

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You should immediately know that it is equal to 1.

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I exclude zero because zero to the zeroth power is usually undefined, or for our purposes you can accept it to be undefined

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and maybe in the future I'll do a video on explaining why its undefined, because it can either-

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There are some good arguments for why it should be zero, because zero to pretty much any other power is zero.

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But then there's a really good argument for why it should be 1, because any other number to the zeroth power is 1.

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So thats why for the most part we like to keep this undefined.

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So with that said, 1 to the zeroth power is just going to be 1. Anything other than zero to the zeroth power is 1.

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So we have 4 times 1 which is 4, minus 2 times 1 which is 2.

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So 4 minus 2 is just equal to 2.

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Now lets do the same thing with n equals 5.

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So it becomes 4 times 5 to the first power,
minus 2 times 5 to the zeroth power.

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Same thing, 5 to the first power is just going to be equal to 5.

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So let me do that in- I want to do that in a different color.

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5 to the first power is just going to be equal to 5.
5 to the zeroth power is just going to be equal to 1.

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So this experssion becomes 4 times 5, minus 2 times 1,

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or 4 times 5 is 20, minus 2 times 1, that is 2.
And this is equal to 18.

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And we're done!