Negative exponents

C

C&D

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Aug 3, 2012
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I get negative exponents, but I don't understand when they put them in parenthesis. Here is a sample
(-15) to the power of -2
and -15 to the power of -2

Wouldn't both be 1/225? Yet that is not my option.
 
C&D said:
I get negative exponents, but I don't understand when they put them in parenthesis. Here is a sample
(-15) to the power of -2
and -15 to the power of -2
Wouldn't both be 1/225? Yet that is not my option.
Click to expand...
NO!
\(\displaystyle (-15)^{-2}=\dfrac{1}{225}\) but \(\displaystyle -15^{-2}=\dfrac{-1}{225}\).
 
C&D said:
I get negative exponents, but I don't understand when they put them in parenthesis. Here is a sample
(-15) to the power of -2
and -15 to the power of -2

Wouldn't both be 1/225? Yet that is not my option.
Click to expand...

To further explain, \(\displaystyle (-15)^{-2}\) means that the entire number, including the negative sign, is being raised to the -2 power. Thus, when you do the reciprocal you must include the negative sign that is being raised to the positive 2 power, i.e. \(\displaystyle \frac{1}{(-15)^2}=\frac{1}{225}\).

However, for \(\displaystyle -15^{-2}\), only the postitive 15 number is being raised to the -2 power. Thus, when you do the reciprocal only the 15 is being raised to the positive 2 power, i.e. \(\displaystyle \frac{1}{-15^2}=\frac{-1}{15^2}=\frac{-1}{225}\).

Make sense?
 
Wow! Thank you!!!!!

It's hard doing summer math without a teacher, but this is great. Thanks for your quick reply!
 
JeffM said:
Notice that (-a)2 follows PEMDAS: we do what is inside the parentheses before exponentiating.

Notice that -a2 also follows PEMDAS: we exponentiate before subtracting.
Click to expand...

I like this response the best because it mentions the Order of Operations. Understanding PEMDAS is sufficient, to be able to distinguish between -15^(-2) and (-15)^(-2).

I will add that when I look at something like -a^2, I do not think of subtraction per se. Rather, I view a leading negative sign as an abbreviation for multiplication by -1.

-a^2 looks like (-1)(a^2), to me.



-15^(-2) looks like (-1)(15^[-2])

Evaluate the power of 15 first, then multiply by -1. That's PEMDAS! :cool:
 
JeffM said:

The distinction between "-" as [an abbreviation] and as an operator can perhaps be deferred to another day.
Click to expand...

Okay! :D

On that day, we could also discuss the following.

2 + -1

I hope that Denis will show up for class that day.

(By the way, if standard keyboards were to have separate keys -- generating separate ASCII symbols -- for the negation sign and subtraction sign, like my TI-89 calculator does, life would be simpler for everybody because the smaller width of the negation sign is clearly distinguishable from the wider width of the subtraction sign. It seems apparent, to me, that not one of the many typewriter inventors gave this issue any thought at all. ;))
 
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