0:00:00.405,0:00:01.456
Let's see if we can apply

0:00:01.456,0:00:04.532
what we know about negative numbers,

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and what we know about exponents

0:00:06.031,0:00:08.879
to apply exponents to negative numbers.

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So let's first think about –

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Let's say we have -3.

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Let's first think about what it means

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to raise it to the 1st power.

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Well that literally means just taking a -3.

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And there's nothing left to multiply it with.

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So this is just going to be equal to -3.

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Now what happens if you were take a -3,

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and we were to raise it to the 2nd power?

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Well that's equivalent to taking 2 -3's,

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so a -3 and a -3, and then multiplying them together.

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What's that going to be?

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Well a negative times a negative is a positive.

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So that is going to be positive 9.

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Let me write this. It's going to be positive 9.

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Well, let's keep going.

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Let's see if there is some type of pattern here.

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Let's take -3 and raise it to the 3rd power.

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What is this going to be equal to?

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Well, we're going to take 3 -3's, [WRITING] –

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and we're going to multiply them together.

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So we're going to multiply them together.

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-3 × -3, we already figured out is positive 9.

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But positive 9 × -3, well that's that's -27.

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And so you might notice a pattern here.

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Whenever we raised raised a negative base

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to an exponent, if we raise it to an odd exponent,

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we are going to get a negative value.

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And that's because when you multiply

0:01:49.402,0:01:50.853
negative numbers an even number of times,

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a negative number times

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a negative number is a positive.

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But then you have one more negative number

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to multiply the result by – which makes it negative.

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And if you take a negative base,

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and you raise it to an even power,

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that's because if you multiply a negative

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times a negative, you're going to get a positive.

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And so when you do it an even number of times,

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doing it a multiple-of-two number of times.

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So the negatives and the negatives all cancel out,

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I guess you could say.

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Or when you take the product of the two negatives,

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you keep getting positives.

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So this right over here is

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going to give you a positive value.

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So there's really nothing new about

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taking powers of negative numbers.

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It's really the same idea.

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And you just really have to remember that

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a negative times a negative is a positive.

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And a negative times a positive is a negative,

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which we already learned from

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multiplying negative numbers.

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Now there's one other thing that I want to clarify –

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because sometimes there might be ambiguity

0:02:50.549,0:02:54.227
if someone writes this.

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Let's say someone writes that.

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And I encourage you to actually pause the video

0:02:58.328,0:03:01.472
and think about with this right over here

0:03:01.472,0:03:03.332
would evaluate to.

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And, if you given a go at that, think about whether

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this should mean something different then that.

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Well this one can be a little bit and big ambiguous

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and if people are strict about order of operations,

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you should really be thinking about the exponent

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before you multiply by this -1.

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You could this is implicitly saying -1 × 2^3.

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So many times, this will usually be interpreted

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as negative 2 to the third power,

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which is equal to -8,

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while this is going to be interpreted

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as -2 to the third power.

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Now that also is equal to -8.

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You might say well what's what's the big deal here?

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Well what if this was what if these were even exponents.

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So what if someone had give myself some more space here.

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What if someone had these to express its -4 or a -4 squared or -4 squared.

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This one clearly evaluates to 16 – positive 16.

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It's a negative 4 times a *4.

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This one could be interpreted as is.

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Especially if you look at order of operations,

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and you do your exponent first,

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this would be interpreted as -4 times 4,

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which would be -16.

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So it's really important to think about this properly.

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And if you want to write the number negative

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if you want the base to be negative 4,

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put parentheses around it and then write the exponent.