In this explainer, we will learn how to use the rules of negative and fractional indices to simplify numerical expressions.
Recall that an exponential is an expression of the form . We refer to the number in such an expression as the base of the exponential and to as the power, exponent, or index. When is a positive whole number, the expression is straightforward to interpret: it means lots of , all multiplied together. For example,
Throughout this explainer, we will interpret expressions of the form when is not a positive whole number, that is, when is negative, a positive fraction, or even a negative fraction.
First of all, let us recall the laws of exponents.
Laws: The Laws of Exponents
For all numbers , , and , we have three laws of exponents:
- The product rule of exponents:
- The quotient rule of exponents:
- The power rule of exponents:
Now, consider the following expression:
We can evaluate this directly as shown:
By canceling common factors in the numerator and denominator, we have that
However, we can also evaluate using the laws of indices: which leads us to the equation
Of course, nothing depends on the base, which is 5 here, and a little experimentation with other bases should convince you that the same argument will yield the equation for any base .
Now, try the same argument again with indices that differ by more than one, for example . By considering a few such examples, we should be able to convince ourselves that a very compelling interpretation of the expression is
Indeed, we will take this as a definition.
Definition: Negative Exponents
For any numbers and , we define
It is worth pausing to note a special case of this definition, namely when is a fraction, say . In this case, we have which, by the definition of negative exponents, is equal to
Since dividing by a fraction is the same thing as multiplying by its reciprocal, we have
That is to say taking a negative power of a fraction has the effect of “flipping” the fraction.
Example 1: Evaluating the Negative Exponent of a Number
Express in the form , where and are integers.
Answer
Recall that for any numbers and , we define
Then, by definition, we have
Thus, we have , , and, for our answer, .
Now, let us turn our attention to fractional exponents. What are we to make, for example, of the expression ?
Well, let us try squaring it:
We can now apply the laws of exponents, specifically the power rule . Thus,
Now, ask yourself the following: What is the equation saying? It is saying that this number, , whatever it is, has the property that when you square it, you get 4. This statement is nothing other than the definition of a square root of 4. Hence, we arrive at the interpretation
Be aware that 4 (like all positive real numbers) has not one but two square roots, namely and . Nothing we have said so far favors one of these over the other as the interpretation of . In order for to be well defined, we follow the convention that denotes the principal (in this case, positive) square root of .
Let us try another expression: but this time, we are going to cube it:
The equation is now telling us that the number (whatever it is) has the property that when you cube it, you get 10. In other words, is a cube root of 10:
A little further experimentation should convince you that a sensible interpretation for the expression is . Indeed, we will take this as a definition.
Definition: Fractional Exponents with Numerator 1
For any numbers and , we define where denotes the principal root of , with the understanding that when is even and is negative, this expression is not defined over the real numbers.
We now have an interpretation for expressions of the form , that is, fractional exponents in which the numerator of the exponent is 1. Once again, we can use the laws of exponents to guide us here. Consider the expression . By the power rule of exponents, , we have to which we can apply the previous definition:
Notice that we can multiply the factors in the exponent (in this example, 2 and ) in either order. That is,
Definition: General Fractional Exponents
For any numbers , , and , we define where denotes the principal root of , with the understanding that when is even and is negative, this expression may not be defined over the real numbers.
The important point here is that in evaluating an expression of the form , the order of the operations of the root and exponent makes no difference. That is, . In practical calculations, you will often find that it is easier to compute the root before the exponent.
Example 2: Evaluating the Fractional Exponent of a Number
Evaluate .
Answer
Recall that for any numbers , , and , we define where denotes the principal root of .
Then, by definition, we have
Now, we have two options for completing the calculation: we can either compute the cube root of or the square of , since the two are equivalent. It is usually easier to compute the root before the exponent, simply because calculating powers of smaller numbers is generally easier than calculating roots of bigger ones. Thus, by squaring first, we have or, by taking the cube root first, we have
Either option gives us the answer .
Be aware that a fractional exponent may sometimes appear in disguised form, that is to say, as a decimal. One should not be afraid of such expressions as
In this situation, simply convert the decimal into a fraction and continue as normal. Thus,
Example 3: Evaluating the Fractional Exponent of a Fraction
Given that , where and are integers, find and .
Answer
Recall that for any numbers , , and , we define where denotes the principal root of .
Then, by definition, we have
To proceed, we use the distributive property of roots over division, that is, the law that for all , , and . So,
Completing the evaluation by raising everything to the power of 4, we have
Thus, , and so our answer is and .
We are now going to combine both negative and fractional exponents in a single calculation. Suppose we are asked to evaluate the expression
The first thing to do is to look at the exponent, which consists of up to three parts: a numerator, a denominator, and a sign (/).
So, in evaluating this example, , we will first take the cube root of 125, we will then square the result, and we will finally take the reciprocal of what we end up with, or in symbols,
Example 4: Evaluating the Negative Fractional Exponent of a Number
Evaluate .
Answer
Recall that for any numbers , , and , we define where denotes the principal root of , and recall that for any numbers and , we define
Looking at the exponent in the expression , we notice two things: a 3 in the denominator, telling us that we need to take a cube root of the base, and a negative sign, telling us to take the reciprocal as a final step. Thus, using the power rule of exponents, ,
The fraction in the exponent is telling us to take a cube root, and finally take the reciprocal, giving us a final answer of .
Our last example will combine everything we have covered in this explainer so far.
Example 5: Evaluating the Negative Fractional Exponent of a Fraction Using Laws of Indices
Evaluate .
Answer
Recall that for any numbers , , and , we define where denotes the principal root of , and recall that for any numbers and , we define
There is more than one way to approach this question. The temptation is perhaps to cube the fraction first and then tackle the exponent of . However, if you do not have a calculator handy, this could be a bit of a challenge. Remember that we can often use the laws of exponents to rearrange the order of calculations, on the basis that in calculations like this, it is usually easier to take roots before powers, saving taking the reciprocal until last. In particular, we will use the power rule of exponents, . So,
Since multiplication is commutative, we can change the order of operations in the exponent:
Applying the power rule, we have
Now, we can take the square root of the innermost parentheses, and apply the power rule of exponents again,
Finally, we take the reciprocal:
Therefore, our final answer is .
Let us finish by recapping a few important concepts from this explainer.
Key Points
- The general formula for negative exponents is given by for any numbers and .
- The general formula for fractional exponents is given by for any numbers , , and .
- When an expression contains some combination of different, possibly negative, possibly fractional exponents, we can often use the index law to rearrange the order of operations. It is usually (though not always) easier to compute roots (that is, exponents of the form ) first and save taking reciprocals (that is, exponents of the form ) until the end.
