Fractional Exponents

Also called "Radicals"

Exponents

The exponent of a number says how many times to use the number in a multiplication.

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"

But what if the exponent is a fraction?

Fractional Exponents: ½

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

Question: What is x^½ ?

Answer:

x^½ = the square root of x

(i.e. x^½ = √x)

Why?

Because if you square x^½ you get: (x^½)² = x¹ = x

Example:

√2 × √2 = 2

Is also:

2^½ × 2^½ = 2¹

To understand that, follow this two-step argument:

First, there is the general rule: (x^m)ⁿ = x^m×n

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

So (x²)³ = x^2×3 = x⁶

Now, let's look at what happens when we square x^½:

(x^½)² = x^½×2 = x¹ = x

When we square x^½ we get x, so x^½ 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 x^¼ ?

(x^¼)⁴ = x^¼×4 = x¹ = x

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

So, x^¼ = The 4th Root of x

General Rule

It worked for ½, it worked with ¼, in fact it works generally:

x^1/n = The n-th Root of x

So we can come up with this:

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

Example: What is 27^1/3 ?

Answer: 27^1/3 = 27 = 3

What About More Complicated Fractions?

What about a fractional exponent like 4^3/2 ? That is really saying to do a cube (3) and a square root (1/2), in any order.

Let me explain.

A fraction (like m/n) can be broken into two parts:

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

So, because m/n = m × (1/n) we can do this:

The order does not matter, so it also works for m/n = (1/n) × m:

And we get this:

A fractional exponent like m/n means:

Do the m-th power, then take the n-th root

OR Take the n-th root and then do the m-th power

Some examples:

Example: What is 4^3/2 ?

4^3/2 = 4^3×(1/2) = √(4³) = √(4×4×4) = √(64) = 8

4^3/2 = 4^(1/2)×3 = (√4)³ = (2)³ = 8

Either way gets the same result.

Example: What is 27^4/3 ?

27^4/3 = 27^4×(1/3) = (27⁴) = (531441) = 81

27^4/3 = 27^(1/3)×4 = (27)⁴ = (3)⁴ = 81

It was certainly easier the 2nd way!

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 see 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

Fractional Exponents

Exponents

Fractional Exponents: ½

Question: What is x½ ?

Why?

Example:

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

(x½)2 = x½×2 = x1 = x