In need of a mnemonic device for rules of exponents

Factual Questions
1

I need a way to remember that:

  1. A negative exponent turns the base into a fraction.
  2. A exponent of 1/n is the same as the nth root.

I always get these two properties confused, to disastrous results.

2

I used to look at:
x[sup]-2[/sup]
and think of the minus sign rotating 90 degrees to become the “1” in:
1 / x[sup]2[/sup]

3

Remember that you wish the property to hold that multiplying powers of a common base results in addition of exponents:

x[sup]m[/sup] * x[sup]n[/sup] = x[sup]m+n[/sup]

both of those facts follow from that

4

Visually, you can base it on whether there’s “already” a fraction on the page. If you see

x[sup]-2[/sup]

there’s a minus sign there, but not a fraction in the form of A/B. So turn it into one:

1 / x[sup]2[/sup]

In the second case, if you see

x[sup]1/7[/sup]

Then there is a fraction on the page, and so you can “get rid of it” or “not create one” or however you want to think about it. Since you know you confuse it with the first case, you know that this one becomes the root (which I don’t know how to represent here!):

[sup]7[/sup])X
Or you can think of the “root” being at the bottom of the tree and below the ground, so your fraction 1/7 is the 7th root.

Or perhaps a “negative” fraction is a “bad” thing, so you want to bury it, and turn it into 1/x[sup]2[/sup]
Yes, I am aware of the username-post combo :slight_smile:

5

Personally, they way you wrote this, 1/n looks (physically) similar to a square root symbol over an ‘n’. Yeah, I know, it’s nth root, not root n, but if I need a mnemonic device, I’d just remember that. Couple that with what xema said and it would work for me.

6

Thanks for the useful suggestions, everyone.

I know that rule, but don’t see how it elucidates fractional or negative exponents. If the sum of two exponents is negative, I still can’t remember whether that should be a root or the reciprocal.

7

Because x[sup]-1[/sup]*x[sup]1[/sup] = x[sup]0[/sup] = 1, so x[sup]-1[/sup] must be 1/x. Similarly, x[sup]1/2[/sup]*x[sup]1/2[/sup] = x[sup]1[/sup], so x[sup]1/2[/sup] must be sqrt(x).

8

Exactly. Understanding is better than blind mnemonics.

Another way to look at it: analogize exponentiation’s relationship to multiplication with multiplication’s relationship to addition:

x * 0 is the additive identity.
Similarly, x ^ 0 is the multiplicative identity.

x * -y is the additive inverse of x * y.
Similarly, x ^ -y is the multiplicative inverse of x ^ y.

x * 1/n is the number you have to add to itself n times to get x.
Similarly, x ^ 1/n is the number you have to multiply by itself n times to get x.

9

Once in a while I get the same momentary dislexia. I usually have to remember that as n increases, x^n always increases (for x>1), and say, "OK, the n-th root of x is greater than one and 1/x is less than one, so the n-th root of x is greater than 1/x^n. And 1/n is greater than -n. So 1/n goes with the n-th root and -n goes with the 1/x^n. "

The whole process goes pretty quickly by now, but I do find myself doing it occasionally.

10

I never had a problem keeping the two ideas separate, but if somehow they’re glued together in your head this may help …

To get from X[sup]3[/sup] to X[sup]2[/sup] you divide by X.
To get from X[sup]2[/sup] to X[sup]1[/sup] you divide by X.
To get from X[sup]1[/sup] to X[sup]0[/sup] you divide by X.
To get from X[sup]0[/sup] to X[sup]-1[/sup] you divide by X.
To get from X[sup]-1[/sup] to X[sup]-2[/sup] you divide by X.

So geting into negative exponents involves repeatedly dividing by X. So your result is a fraction with your X in the denominator. X[sup]-n[/sup] == 1 / X[sup]+n[/sup]

And since now you know that the negative exponent pattern produces 1/X-style results, then by process of elimination the fractional exponent pattern must produce nth-root style results. X[sup]1/n[/sup] == n[sup]th[/sup] root of X

Viola!!
Here’s an alternative approach if you do anything with computer programming, even just spreadsheets …

To take the square root, you write a formula like


result = X^0.5

Well it’s pretty obvious that 0.5 = 1/2, so the fractional exponent must be the pattern for taking n[sup]th[/sup] roots.

And so by process of elimination, negative exponents must be the pattern for reciprocals. So just remember “X^0.5” == “take square root”.
One of these two approaches ought to click for you. The first one even adds some understanding.

11

These are both very useful, thanks. It’s always better if I can derive the rule than just remembering something, but I was having trouble getting from point A to B.