If a number is raised to a negative power, is the answer always negative?

Question

If a number is raised to a negative power, is the answer always negative? For example, if I have 2 raised to the negative 2 power is the answer -4 or 4? Or if I have 2 raised to the -3 power, is the result 8 or -8? I'm a little rusty on my Algebra/Calcus. Please help...

Answer 1

You have 14 answers, so by now you know what it really means: it means 1 over that same number but now with a positive exponent.
I'm adding my response to your collection because I want to make sure you know WHY this is so.

In the laws for exponents, we are told that under multiplication, with like bases, you add the exponents to get the result
Thus, (x^3)(x^2) = x^5
This makes sense because x^3 is (x)(x)(x) and
x^2 is (x)(x), so (x^3)(x^2)=(x)(x)(x)(x)(x), or x^5

Similarly, in division, x^5/x^2 =(x)(x)(x)(x)(x)/(x)(x)=x^3
The rule here becomes "subtract the exponents when dividing". The bases must, of course, be the same.

But look what we get if the problem is x^2/x^5 !
It is (x)(x)/(x)(x)(x)(x)(x), or x^-3 under the subtraction rules we just made for division.
We know that (x)(x)/ (x)(x)(x)(x)(x) is = 1/(x)(x)(x), or
1/ x^3. Conclusion?
x^-3 and 1/ x^3 mean the same thing!

I'd like to go just a bit farther with my words.
x^2/ x^2 =1, but under our division rules we get
x^2-2, or x^0. Conclusion? x^0 =1

Every teacher of math that was or will be born loves to throw the messiest problem you can imagine that has the exponent 0 in it. Example: {13xy^2(k^-6}^0.
As soon as you see that 0, write down 1 as your answer and enjoy a good night's sleep.
Let B = {13xy^2(k^-6)}
Then problem becomes B^0, which =1

Good luck with your studies.

Answer 2

Raising a number to a negative power does not necessarily mean it is negative. The negative means that you are raising the reciprocal to that power. In other words 2 raised to the negative 2 power is the same as 1/2 raised to the positive 2 power. So the answer is 1/4. Likewise, 2 raised to the -3 power is 1/2 raised to the +3 power which is equal to 1/8.
Note that a positive number raised to any power (whether positive or negative) will always be positive.

Hope that helps.

Answer 3

This Site Might Help You.

RE:
If a number is raised to a negative power, is the answer always negative?
If a number is raised to a negative power, is the answer always negative? For example, if I have 2 raised to the negative 2 power is the answer -4 or 4? Or if I have 2 raised to the -3 power, is the result 8 or -8? I'm a little rusty on my Algebra/Calcus. Please help...

Answer 4

No. If you raise 2 to the -2 power, it will be 1/4.
If you raise 2 to the -3 power, it will be 1/8.

e.g.
Raising a nonzero number to the −1 power produces its reciprocal.


Thus:


Raising 0 to a negative power would imply division by 0, and so is undefined.

A negative integer exponent can also be seen as repeated division by the base. Thus

Answer 5

Certainly not a negative sign when used to modify an exponent indicates the number gets smaller not bigger

example 10exp 3 is 1000 10 exp-3 is 1/1000 or 0.001

.001 is still greater than 0

numbers using exponents are only negative in value if
1/the initial integer is negative in value and
2/ the exponent is an odd number
-10 exp 2 =+100 -10 exp3 = -1000

Mike

Answer 6

the answer is not negative. A negative power creates the multiplicative inverse or reciprocal of the number.

2^2 = 4 so 2^-2 = 1/4

3^2 = 9 so 3^-2 = 1/9

remember scientific notation ? 10^-3 = .001, etc ?

Answer 7

No, it will never be negative. Raising a number to a negative is the same thing as saying 1 divided by the number and raising that to the same positive. For instance:

a^(-b) = 1/ (a^b) or (1/a)^b

So, it can never come out negative, unless the first number is negative to begin with.

Answer 8

no, any number has a negative power will become a division. for example, 2 to the power of negative 2 is 1 over 2 the the power of 2

2^(-2) = 1 / (2^2)
=1/4 or 0.25

likewise, 2 ^(-3) is 1/8 or 0.125

Answer 9

A negative power is the same as putting 1 over the positive power.

so, given your examples:

2^(-2) = 1/4
2^(-3) = 1/8

Answer 10

A number raised to a negative power is the same as the reciprocal of the number being raised to a positive power. In general,

x^(-n) = 1/(x^n)