In this explainer, we will learn how to simplify fractional indices.

When we talk about expressions that contain rational exponents, we are referring to any expression in the form where the exponent is a fraction.

In order to introduce rational exponents, let us first recall the product rule of exponents, which states that

If we then consider the expression the product rule of exponents tells us that this must be equivalent to which simplifies to . Looking back, this means that must be equal to a number, say , such that when we multiply by itself we get an answer of 9. That is,

This means that and that

This can be generalized for any ; that is,

What about exponents of a third or a quarter? Well, we can take a very similar approach as we did before: if we consider the expression using the product rule of exponents this simplifies to 8. As before, this means that must be equal to a number, let us call it this time, such that when we multiply by itself three times we get an answer of 8. That is, , which means that and that

Using a similar construction, we can conclude that

Following this discussion, we can state the following rule.

### Key Rule: Rational Exponents with a Numerator of One

For any variable , we have that

Let us look at an example.

### Example 1: Evaluating Rational Exponents

Evaluate .

### Answer

If we start by recalling the rule for rational exponents with a numerator of one, we have that

Using this rule, we can rewrite our expression as follows:

We then need to identify the cube root of 64. This is the number that, when multiplied by itself three times, equals 64. This number is 4 as .

What happens then if the numerator of our rational exponent is not equal to one? To answer this, we need to recall the power rule of exponents:

If we consider the expression , we can rewrite this as . The power rule of exponents tells us that this is the same as which means that this is the same as

Note here that we could also rewrite as , which, again by the power rule, can be rewritten as

This is the same as

The first form is much easier to evaluate; we have that , and , so

Using what we have found above, we can now state the more general rule for rational exponents.

### Key Rule: Rational Exponents

For any variable , we have that

Note here that we have stated both ways in which you can rewrite rational exponent expressions, but the first form is generally easier to use when evaluating numerical expressions.

Let us now have a look at a few examples where we use the rule for rational exponents to rewrite, simplify, or evaluate rational exponents.

### Example 2: Rewriting Rational Exponents

Express in the form .

### Answer

If we start by recalling the rule for rational exponents, we have that

In the question, we have been asked to write the expression in the second form with the exponent inside the radical. We can, therefore, rewrite the expression as follows:

In this example, we have demonstrated how we can use the rule of rational exponents to convert from an exponent expression to a radical expression. We can also use the rule in the reverse direction, which we will demonstrate now.

### Example 3: Rewriting Rational Exponents

Express in the form .

### Answer

If we start by recalling the rule for rational exponents, we have that

Comparing this with the questions, we need to use the fact that

Directly comparing this with our expression, we can see that this can be rewritten as

Having the fluency to transfer between the two different ways of writing rational exponent expressions is a vital skill in our ability to evaluate and simplify these types of expressions. Let us now look at a couple of examples where we are required to evaluate various expressions.

### Example 4: Evaluating Rational Exponents

Evaluate .

### Answer

If we start by recalling the rule for rational exponents, we have that

We can use this to rewrite our expression as follows:

At this point, we can evaluate the expression. If you do not know your fourth roots, it is worth pointing out that you can calculate these by square rooting twice. In this case, the square root of 16 is 4, and the square root of 4 is 2. Therefore, the fourth root of 16 is 2. Using this, we can simplify our expression as follows:

Let us finish by looking at an example where we are required to simplify an expression containing more than one rational exponent.

### Example 5: Evaluating Rational Exponents

Calculate .

### Answer

You may notice with this question that each of the fractions is not presented in its simplest form. We can make the task of evaluating the expressions easier by first reducing the fractions to simplest form, that is,

At this point, we can distribute the exponent across the numerator and denominator of each of the fractions to get

Now, let us recall the rule for rational exponents which tells us that

We can use this to rewrite our expression:

We can now evaluate all of the radicals to get and now the exponents to get

Finally, we can multiply the fractions by cross canceling:

### Key Points

- The rules of exponents can be extended to fractional exponents. Both the product and
power rules extend to fractional exponents:
- ; for example, ;
- ; for example, .

- Fractional exponents can be converted to rational expressions and vice versa using the
following rules:
- ; for example, ;
- ; for example, ;
- ; for example, .