Exponents and negative numbers

Why do I get a negative answer even though the exponent is even and the other exponent is odd? Doesn't the exponent being odd or even determine if the answer is negative or positive?

Here are two examples:

−(3/2)2 = −2 1/4.
−(3/2)3 = −3 3/8.

This is really a matter of notation.

We have agreed that −(3/2)2 is not the same as (-3/2)2.

If you want to calculate -3/2 to the second power, it is written so that -3/2 is inside parenthesis, like this: (-3/2)2.

The answer to this is indeed positive: it is (-3/2) × (-3/2) = 9/4 = 2 1/4.

However, if the minus sign is outside the parenthesis, it means the opposite number of whatever it's in front of.

So, −(3/2)2 means: "First calculate (3/2)2, then take the opposite of that."
First you calculate (3/2)2 = 9/4, and its opposite is -9/4.

In other words, the location of the parenthesis makes a huge difference.

With exponent 3, we get:

−(3/2)3 = −(3/2 × 3/2 × 3/2) = −27/8.

(-3/2)3 = (-3/2) × (-3/2) × (-3/2) = -27/8 or -3 3/8.

So in this case, these two expressions have the same value.

An advanced student may notice:

If n is even, and x is any real number, then

(-x)n = xn ; in other words it won't matter if you calculate the power using x or using negative x - the result is the same.

And if n is odd, then

(-x)n = −xn ; in other words, it's like you can "pull the minus out" from inside those parenthesis and put it out front.

Comments

Anonymous said…
Nice post! Another useful lesson here is that since, for example, -3 = -1 * 3, we have that

(-3)^n = (-1)^n * 3^n. Especially for the advanced students, this can give a great explanation for why the even/odd rule works the way it does.

Cool stuff!
3:02 PM, February 14, 2009
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