1. Negative Integers and exponents.

Ok, so my homework was to figure out how negative integers are affected by exponents. The questions were:

-3^2 = __
-3^3 = __
-3^4 = __
-3^5 = __

So I got positive answers for all of these questions. Then I noticed one last problem at the end of my assignment:

In working with powers having a negative base:
--if the exponent is odd, the sign on the answer is ______________.
--if the exponent is even the sign on the answer is ______________.

So my question is: Why does the odd/even thing make a difference as to the sign of the answer? I know what the actual answers to the last questions ARE, thanks to a bit of googling, I just don't understand the hows and whys.

Thanks

2. Originally Posted by MMM
Ok, so my homework was to figure out how negative integers are affected by exponents. The questions were:

-3^2 = __
-3^3 = __
-3^4 = __
-3^5 = __

So I got positive answers for all of these questions. Then I noticed one last problem at the end of my assignment:

In working with powers having a negative base:
--if the exponent is odd, the sign on the answer is ______________.
--if the exponent is even the sign on the answer is ______________.

So my question is: Why does the odd/even thing make a difference as to the sign of the answer? I know what the actual answers to the last questions ARE, thanks to a bit of googling, I just don't understand the hows and whys.

Thanks
$-3^2 = -3 \times -3 = + 9$

$-3^3 = -3 \times -3 \times -3 = +9 \times -3 = - 27$

Remember that a positive times a negative is a NEGATIVE.

And that a negative times a negative is a POSITIVE.

So in the case of something like:

$-3^3 = -3^2 \times -3$

$Notice \ that \ -3^2 = +9$

But multiplying that by -3 is a positive times a negative.

So that will give us a negative, and 9 x 3 = 27

So that gives -27

---

So what we can conclude is that when the exponent is an even number, we will always have a positive answer.

If the exponent is odd, the answer depends on what we started out with.

-3 or -4 or -5, etc to any odd power will ALWAYS give a negative answer.

But any value to an even power will ALWAYS give a positive answer.

A positive value to an odd power will give a postive answer.

A positive value to any power(even or odd, positive or negative) will ALWAYS be positive.

3. Originally Posted by MMM
Ok, so my homework was to figure out how negative integers are affected by exponents. The questions were:

-3^2 = __
-3^3 = __
-3^4 = __
-3^5 = __

So I got positive answers for all of these questions. Then I noticed one last problem at the end of my assignment:

In working with powers having a negative base:
--if the exponent is odd, the sign on the answer is ______________.
--if the exponent is even the sign on the answer is ______________.

So my question is: Why does the odd/even thing make a difference as to the sign of the answer? I know what the actual answers to the last questions ARE, thanks to a bit of googling, I just don't understand the hows and whys.

Thanks
Hi MMM,

Do a little experimenting!

$(-3)^1=-3$

$(-3)^2 = (-3)(-3) = 9$
Sign diagram:
$- \times - = +$

$(-3)^3 = (-3)(-3)(-3) = (-3)^2(-3)= 9(-3)=-27$
Sign diagram:
$- \times - \times - = (- \times -) \times -= + \times - = -$

$(-3)^4 = (-3)(-3)(-3)(-3) = (-3)^2 (-3)^2 = (9)(9) = 81$
Sign diagram:
$- \times - \times - \times - = (- \times -) \times (- \times -) = + \times + = +$

And this pattern keep alternating no matter how high powered you go!

Also, there is something very important about exponents:
$-3^2 \neq (-3)^2$
$-3^4 \neq (-3)^4$
Why? Because $-3^2 = -(3^2)=-9$, but $(-3)^2 = 9$.
In the first case we are applying the negative After we apply the power. In the second case, we are including the negative sign in the number that is to be raised to a power. There is a big difference.
Try to convince yourself by trying some examples that
$-3^n \neq (-3)^n$
If n is even, and
$-3^n = (-3)^n$
If n is odd.

4. Originally Posted by MMM
Ok, so my homework was to figure out how negative integers are affected by exponents. The questions were:

-3^2 = __
-3^3 = __
-3^4 = __
-3^5 = __
Never ever use an expression like -3^3, there are conventions that say that
this means -(3^2), but they are often violated. so don't do it. Make what
you want such an expression to mean explicit by using brackets.

So always write -(3^2) or (-3)^2.

ZB

5. Originally Posted by Constatine11
Never ever use an expression like -3^3, there are conventions that say that
this means -(3^2), but they are often violated. so don't do it. Make what
you want such an expression to mean explicit by using brackets.

So always write -(3^2) or (-3)^2.

ZB
Actually, by order of operations, squaring comes before negation, so by definition
$-3^2 = -(3^2) = -9$

-Dan

6. Originally Posted by MMM
...

So my question is: Why does the odd/even thing make a difference as to the sign of the answer? I know what the actual answers to the last questions ARE, thanks to a bit of googling, I just don't understand the hows and whys.

Thanks
Hello,

every number consists of a sign and an absolute value. Let the number be x. Then you have:

$x = sgn(x) \cdot | x |$

sgn(x) has only 2 values:

$sgn(x)=\left \{ \begin{array}{lcr}+1 & \text{ if } & x>0 \\ -1 & \text{ if } & x<0 \end{array} \right.$

Now consider a negative number taken to the power of $n \in \mathbb{N}$

$x^n=(-1 \cdot |x|)^n = (-1)^n \cdot (| x |)^n$

$(-1)^n=\left \{ \begin{array}{lcr}+1 & \text{ if } & \text{n is even} \\ -1 & \text{ if } & \text{n is odd} \end{array} \right.$

That means: If you take a negative number to an odd power(?) the result will be negative.

7. Originally Posted by topsquark
Actually, by order of operations, squaring comes before negation, so by definition
$-3^2 = -(3^2) = -9$

-Dan
Try telling that to Excel or Open Office

Or let me make myself clearer. Spread sheets treat unary "-" as higher precedence than "^", but
binary "-" as of lower precedence.

So -3^2 is 9, while 0-3^2 is -9 according to Excel.

Hence my warning, never leave out the brackets and rely on the conventional operator precedence
in a calculation.

Excel is one of the most common tools for a lot of tasks outside maths, and it is also the most
common source of buggy calculations/applications.

ZB

8. Hi there,

I'm pretty sure my instructor intended for all of the problems to be in the (-3)^2 format. Anyway, I think I understand this now. I guess my problem was in going through the multiplication too quickly (e.g., 3x3x3 on a calculator :P) rather than thinking the problem through logically.

Thanks a bunch