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## 8th grade

### Course: 8th grade > Unit 1

Lesson 5: Exponents with negative bases- Exponents with negative bases
- The 0 & 1st power
- Exponents with integer bases
- Exponents with negative fractional bases
- Even & odd numbers of negatives
- 1 and -1 to different powers
- Sign of expressions challenge problems
- Signs of expressions challenge
- Powers of zero

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# Exponents with negative bases

Learn to what we know about negative numbers to determine how negative bases with exponents are affected and what patterns develop. Also learn how order of operations affect the pattern. Created by Sal Khan.

## Want to join the conversation?

- Is there a real life situation where an exponent with a negative base (-x^3) would be used? Could you give me a word problem using it?(17 votes)
- How come I'm practicing 8th grade math and it states on one of the problems that negative 7 squared is negative 49. I thought it was positive. Apparently, the problem stated in the hints that the negative sign is not part of the base of the exponent and therefore, is a negative number. I did not quite understand this concept.(9 votes)
- -7^2 is -49, because of the order of operation. You have to do exponents before doing the negative sign(the coefficient, which is -1(7); multiplication). However, (-7)^2 = 49, because brackets go first.

This refers to3:11.(5 votes)

- at1:03I get the exponent but I don't get negative.(4 votes)
- Hello,

There's a rule which state (-)*(-) [ minus times minus ] = a + , for more information: https://www.khanacademy.org/math/arithmetic/arith-review-negative-numbers#arith-review-mult-divide-negatives

Now khan just made a (-3)*(-3)*(-3) We need to multiply 3 times -3

-3*-3 = +9

+9*-3 = -27

Hope I helped , IF not the link sure will.(5 votes)

- Why do we need negative exponents? If so how do we use negative exponents?(3 votes)
- Negative exponents are useful for representing things that are minuscule, like bacteria, or human cells. To get these values, you would use scientific notation.(4 votes)

- If I see -4^2 with no parenthesis, can I safely assume that it would be positive 16?(2 votes)
- No - by the order of operations, exponents come before subtraction (which includes negative signs). The exponent is applied before the negative symbol, yielding -16.(3 votes)

- (-2) ^2

So for this, it would just be - 2 ✕ - 2 ? Just checking!

Thanks!(3 votes) - How you can tell if a negative number raised to a power is going to be positive or negative without working it out?(1 vote)
- The answer is surprisingly simple! A negative number raised to an odd power is always negative, and a negative number raised to an even power is always positive.

For example, (-6)^11 is negative and (-6)^12 is positive.

(Note well: when writing a negative number to a power, parentheses should be placed around the negative number. Otherwise, the negative sign would be applied after the power is taken. This makes a difference if the exponent is even. So raising the quantity -6 to the 12th power is**not**the same as computing -6^12. Note that -6^12 would be negative instead of positive.)

Have a blessed, wonderful day!(4 votes)

- is there an easier way to make the equation -1 to the power of 823? i would love to find an easier solution because my math teacher just isn't getting through to me on how to do this type of math.(2 votes)
- 823 is an odd value. Hence (-1)^odd value is negative which -1.(2 votes)

- what if it says 2(-11) ? -22 would be my answer(2 votes)
- Indeed !

Be aware of 2-11 which can be written as 2+(-11) which is different ( = -9 ).(2 votes)

- When^ dividing or multiplying powers with the same base, you just add or subtract the exponents. If I had an example -6^3 divided by (-6)^2, or 6^3 divided by (-6)^2, can I subtract the exponents? Because aren't you only allowed to combine numbers with the same bases- and in this case, we've got 2 bases- our first base is a positive six, and our second base is a negative 6. So are they allowed to be combined? Also, would my answer be positive or negative? Thank you(2 votes)
- In your first example: (-6)^3 / (-6)^2 Yes, you can subtract the exponents since the numerator & the denominator have the same base (-6), and your answer would be (-6)^(3-2) = (-6)^1 = -6

However, in your second example you can't subtract the exponents unless you make the bases the same, and to do this you have to know this little trick which is (a x b)^n = (a^n) x (b^n)

For example: (3 x 4)^2 = (3^2) x (4^2) = 9 x 16 = 144

Now, let's do your example:`6^3 / (-6)^2`

Let's make the base in the denominator positive by splitting (-6) into (-1 x 6)`= 6^3 / (-1 x 6)^2`

Let's use the trick we just learned: (-1 x 6)^2 = (-1)^2 x 6^2`=6^3 / ( (-1)^2 x 6^2 )`

But -1 x -1 = 1 So (-1)^2 = 1`=6^3 / ( 1 x 6^2)`

But 1 multiplied by any number is the same number`=6^3 / 6^2`

And now the bases are the same, so we can subtract the exponents`=6^(3-2) = 6^1 = 6`

And, you can do this trick whenever the bases have the same value, but one of them is positive and the other is negative.(1 vote)

## Video transcript

Let's see if we can apply what we know about negative numbers, and what we know about exponents to apply exponents to negative numbers. So let's first think about – Let's say we have -3. Let's first think about what it means to raise it to the 1st power. Well that literally means just taking a -3. And there's nothing left to multiply it with. So this is just going to be equal to -3. Now what happens if you were take a -3, and we were to raise it to the 2nd power? Well that's equivalent to taking 2 -3's, so a -3 and a -3, and then multiplying them together. What's that going to be? Well a negative times a negative is a positive. So that is going to be positive 9. Let me write this. It's going to be positive 9. Well, let's keep going. Let's see if there is some type of pattern here. Let's take -3 and raise it to the 3rd power. What is this going to be equal to? Well, we're going to take 3 -3's, [WRITING] – and we're going to multiply them together. So we're going to multiply them together. -3 × -3, we already figured out is positive 9. But positive 9 × -3, well that's that's -27. And so you might notice a pattern here. Whenever we raised raised a negative base to an exponent, if we raise it to an odd exponent, we are going to get a negative value. And that's because when you multiply negative numbers an even number of times, a negative number times a negative number is a positive. But then you have one more negative number to multiply the result by – which makes it negative. And if you take a negative base, and you raise it to an even power, that's because if you multiply a negative times a negative, you're going to get a positive. And so when you do it an even number of times, doing it a multiple-of-two number of times. So the negatives and the negatives all cancel out, I guess you could say. Or when you take the product of the two negatives, you keep getting positives. So this right over here is going to give you a positive value. So there's really nothing new about taking powers of negative numbers. It's really the same idea. And you just really have to remember that a negative times a negative is a positive. And a negative times a positive is a negative, which we already learned from multiplying negative numbers. Now there's one other thing that I want to clarify – because sometimes there might be ambiguity if someone writes this. Let's say someone writes that. And I encourage you to actually pause the video and think about with this right over here would evaluate to. And, if you given a go at that, think about whether this should mean something different then that. Well this one can be a little bit and big ambiguous and if people are strict about order of operations, you should really be thinking about the exponent before you multiply by this -1. You could this is implicitly saying -1 × 2^3. So many times, this will usually be interpreted as negative 2 to the third power, which is equal to -8, while this is going to be interpreted as -2 to the third power. Now that also is equal to -8. You might say well what's what's the big deal here? Well what if this was what if these were even exponents. So what if someone had give myself some more space here. What if someone had these to express its -4 or a -4 squared or -4 squared. This one clearly evaluates to 16 – positive 16. It's a negative 4 times a *4. This one could be interpreted as is. Especially if you look at order of operations, and you do your exponent first, this would be interpreted as -4 times 4, which would be -16. So it's really important to think about this properly. And if you want to write the number negative if you want the base to be negative 4, put parentheses around it and then write the exponent.