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Fractional Exponents

Also called "Radicals"

Exponents

8 to the Power 2

The exponent of a number says how many times to multiply the number.

In this example: 82 = 8 × 8 = 64

  • In words: 82 could be called "8 to the second power", "8 to the power 2" or simply "8 squared"

Fractional Exponents: ½

In the example above, the exponent was "2", but what if it were "½" ? How does that work?

Question: What is x½ ?

Answer: x½ = the square root of x   (ie x½ = √x)

Why?

Because if you square x½ you get: (x½)2 = x1 = x

To understand that, follow this two-step argument:

1

First, there is the general rule: (xm)n = xm×n
(Because you multiply x "m" times, then do that "n" times, for a total of m×n)

Example: (x2)3 = (xx)3 = (xx)(xx)(xx) = xxxxxx = x6

So (x2)3 = x2×3 = x6

2

Now, let's look at what happens when we square x½:

(x½)2 = x½×2 = x1 = x

When we square x½ we get x, so x½ must be the square root of x

Try Another Fraction

Let us try that again, but with an exponent of one-quarter (1/4):

What is x¼ ?

(x¼)4 = x¼×4 = x1 = x

So, what value can be multiplied 4 times to get x? Answer: The fourth root of x.

So, x¼ = The 4th Root of x

General Rule

In fact we can come up with a general rule:

A fractional exponent like 1/n means to take the n-th root:

Example: What is 271/3 ?

Answer: 271/3 = 27 = 3

What About More Complicated Fractions?

More complicated fractions can be broken into two parts:

  • a whole number part, and
  • a fraction (1/n style) part

. To understand this, just remember that m/n = m × (1/n):

So, we get this:

A fractional exponent like m/n means to do the m-th power, then take the n-th root

Example: What is 43/2 ?

Answer: 43/2 = 43×(1/2) = √(43) = √(4×4×4) = √(64) = 8

Now ... Play With The Graph!

See how smoothly the curve changes when you play with the fractions in this animation, this shows you that this idea of fractional exponents fits together nicely. Things to try:

  • Start with m=1 and n=1, then slowly increase n so that you can se 1/2, 1/3 and 1/4
  • Then try m=2 and slide n up and down to see fractions like 2/3 etc
  • Now try to make the exponent -1
  • Lastly try increasing m, then reducing n, then reducing m, then increasing n: the curve should go around and around