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### Course: Grade 7 (Virginia)>Unit 8

Lesson 2: 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?
• i swear youll never use this.
• 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.
• 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.
• How you can tell if a negative number raised to a power is going to be positive or negative without working it out?
• 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!
• 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.
• 823 is an odd value. Hence (-1)^odd value is negative which -1.
• what if it says 2(-11) ? -22 would be my answer
• Indeed !

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

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

Thanks!
• 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
• 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

`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`