Exponents & Logarithms: Fractional Exponents

Fractional Exponents

Also called "Radicals" or "Rational Exponents"

Whole Number Exponents

First, let us look at whole number 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"

Another example: 5³ =5 × 5 × 5 = 125

Fractional Exponents

But what if the exponent is a fraction?

An exponent of 1/2 is actually square root And an exponent of 1/3 is cube root An exponent of 1/4 is 4th root And so on!

Why?

Let's see why in an example.
First, the Laws of Exponents tell us how to handle exponents when we multiply:

Example: x²x² = (xx)(xx) = xxxx = x⁴

Which shows that x²x² = x⁽²⁺²⁾ = x⁴

So let us try that with fractional exponents:

Example: What is 9^½ × 9^½ ?

9^½ × 9^½ = 9^(½+½) = 9⁽¹⁾ = 9

So 9^½ times itself gives 9.
What do we call a number that, when multiplied by itself gives another number? The square root!
See:

√9 × √9 = 9

So 9^½ is the same as √9

Try Another Fraction

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

Example: x^¼

x^¼ × x^¼ × x^¼ × x^¼ = x^{(¼+¼+¼+¼)} = x⁽¹⁾ = x

So x^¼, when used 4 times in a multiplication gives x, and so x^¼ is 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

or

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

or

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

Posted in: Exponent & Logarithms, Fractions