Explainer: Fractional Exponents

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 𝑥, where the exponent is a fraction.

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

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

This means that 𝑥=9 and that 9=9=3.

This can be generalized for any 𝑥; that is, 𝑥=𝑥.

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 8×8×8, using the product rule of exponents this simplifies to 8. As before, this means that 8 must be equal to a number, let us call it 𝑦 this time, such that when we multiply 𝑦 by itself three times we get an answer of 8. That is, 𝑦=8, which means that 𝑦=8 and that 8=8=2.

Using a similar construction, we can conclude that 𝑥=𝑥.

Following this discussion, we can state the following rule.

Key Rule: Rational Exponents with a Numerator of One

For any variable 𝑥, we have that 𝑥=𝑥.

Let us look at an example.

Example 1: Evaluating Rational Exponents

Evaluate 64.

Answer

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

Using this rule, we can rewrite our expression as follows: 64=64.

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 4×4×4=64.

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: (𝑥)=𝑥.

If we consider the expression 27, we can rewrite this as 27×. The power rule of exponents tells us that this is the same as 27, which means that this is the same as 27.

Note here that we could also rewrite 27 as 27×, which, again by the power rule, can be rewritten as 27.

This is the same as 27.

The first form is much easier to evaluate; we have that 27=3, and 3=9, so 27=9.

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

Key Rule: Rational Exponents

For any variable 𝑥, we have that 𝑥=𝑥=𝑥.

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 𝑥 in the form 𝑎.

Answer

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

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: 𝑥=𝑥.

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 𝑥 in the form 𝑥.

Answer

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

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

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

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 16.

Answer

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

We can use this to rewrite our expression as follows: 16=16.

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: 16=16=2=8.

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 1636×64512.

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, 49×18.

At this point, we can distribute the exponent across the numerator and denominator of each of the fractions to get 49×18.

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

We can use this to rewrite our expression: 49×18.

We can now evaluate all of the radicals to get 23×12 and now the exponents to get 827×12.

Finally, we can multiply the fractions by cross canceling: 827×12=427.

Key Points

  1. The rules of exponents can be extended to fractional exponents. Both the product and power rules extend to fractional exponents:
    • 𝑥×𝑥=𝑥; for example, 2×2=2;
    • (𝑥)=𝑥; for example, 2=2.
  2. Fractional exponents can be converted to rational expressions and vice versa using the following rules:
    • 𝑥=𝑥; for example, 125=125=5;
    • 𝑥𝑥; for example, 16=16=2=8;
    • 𝑥𝑦=𝑥𝑦; for example, 2764=2764=34=81256.

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