Fractional Exponents

A fractional exponent is an alternate notation for expressing powers and roots together. Fractional Exponents are also known as Radicals or Rational Exponents.

Whole Number Exponents

First, look at whole number exponents:

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

Examples

62 = 6 × 6 = 36

43 = 4 × 4 × 4 = 64

In words:

62 could be called “6 to the second power”, or “6 to the power 2” or simply “6 squared”

Fractional Exponents

Remember that when aa is a positive real number, both of these equations are true:

Exponents

If the exponent is a fraction?

When you have a fractional exponent, the numerator is the power and the denominator is the root. In the variable. So:

Fractional Exponents

Where:

  • x is a real number
  • a and b are positive real numbers
  • a is the power
  • b is the root

Exponent of FractionExample
An exponent of ½ is square rootFractional Exponents Example 1
An exponent of ⅓ is cube rootFractional Exponents Example 2
An exponent of ¼ is 4th rootFractional Exponents Example 3
And so on!etc

General Rule

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

x1/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:

Fractional Exponents
Example

What is 81/3 ?

Answer:

81/3 = 3√8 = 2

More Complicated Fractions

What about a fractional exponent like 93/2 ?

That is really saying to do a cube (3) and a square root (1/2), in any order.

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

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

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

Fractional Exponents m n

And we get this:

A fractional exponent like m/n means:

 In SymbolsExample
Do the m-th power, then take the n-th rootFractional Exponents m n 193/2 = 2√(93) = 2√729 = 27
Or Take the n-th root and then do the m-th powerFractional Exponents m n 293/2 = (2√9)3 = 33 = 27

Note

If they give you x3/6, then x had better not be negative, because x3 would still be negative, and you would be trying to take the sixth root of a negative number.

If they give you x4/6, then a negative x becomes positive (because of the fourth power) and then it is sixth-rooted, so by reducing the fractional power it becomes |x|2/3.

But, if they give you something like x4/5, then you don’t need to care whether x is positive or negative, because a fifth root doesn’t have any problem with negatives.

Learn More

Exponents

Negative Exponents

Simplifying Square Roots

Cube Numbers and Cube Roots

Algebra Index

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