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Lesson 5: Exponents with negative bases

# 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?
• a real-life situation where you might use a positive exponent with a negative base. Suppose you're a scientist studying the behavior of a certain type of bacteria. You notice that every hour, the number of bacteria decreases by a factor of 5. After 4 hours, how many times less bacteria will there be compared to the original amount? In this case, you can use the expression -5^4 to calculate the change in the number of bacteria. The base -5 represents the factor by which the number of bacteria decreases every hour, and the exponent 4 represents the number of hours. According to the order of operations, you should calculate 5^4 first, then multiply the result by -1. So, -5^4 = -1*5*5*5*5 = -625. This means that after 4 hours, there will be 625 times less bacteria than the original amount.
• 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.
• -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 to .
• at I get the exponent but I don't get negative.
• Why do we need negative exponents? If so how do we use negative exponents?
• Negative exponents are useful for representing things that are minuscule, like bacteria, or human cells. To get these values, you would use scientific notation.
• i have been watching this video and trying to solve problems about this topic, but for some reason i keep getting them wrong. i don't understand the negative base. can anyone help me please?
• The problem you are having is that multiplying a negative number with a negative number is a positive. So, when you have a negative base, it will always be positive. For example:

(-8)^2
To solve this, all you have to do is multiply -8 by -8, or -8 x -8. As you can see, multiplying a negative number by a negative equals a positive.

Hope this helps you!
• my mind is blown, I mean like who would think that if you do -4 to the second power and you do (-4)to the second power you get different answers!
• The reason for this problem is the order of how you do it. The first one is 4 squared first, and the other is doing minus 4 first.

I agree with you. It made me dizzy when I had first done it!
(1 vote)
• (-2) ^2

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

Thanks!
• No. I believe that what you thought was right, but you wrote it the wrong way. It needs to be (-3)*(-3), because we are multiplying negative numbers.
• A solar powered water system has a solar panel containg 8 rows of moduels . How many modules are there in each row if there are 8 raise to the power 8 modules in all
• Ok, lets say you had 8^2 , that would give you 64 modules right. but if you had 8 rows you would divide that number by 8, correct? That would give you 8 modules in 8 rows. Remember multiplication and division are the opposite of each other. So if you had a total of 8^8 modules in all, and you divided that number by 8 it would give you 8^7. If you divided 8^7 by 8 it would give you 8^6, and so on and so on. So if you have 8^8 modules in all, and you want to know how many in each row, and you have 8 rows you would divide 8^8 by 8 which would give you 8^7 which equals 2,097,152. Hope this helps :)
• On a calculator -3^2 equals -9 but (-3)^2 equals 9. Why is that?
(1 vote)
• Without parentheses, the exponent is applied before the negative sign.

However, if the negative number is in parentheses, then the exponent is applied to the entire negative number.

In the expression -3^2, 3^2 is calculated to give 3*3 = 9, then the negative sign is attached to give a final answer of -9. It helps to think of attaching the negative sign as multiplying by -1, and to recall that exponents are performed before multiplication according to PEMDAS.

However, in the expression (-3)^2, the exponent 2 is applied to the entire number -3, so (-3)^2 = (-3)*(-3) = 9.

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