In this explainer, we will learn how to simplify and evaluate expressions involving zero and negative indices.
Let us first recall what a power is.
Definition: Powers and Exponential Form
Powers are numbers resulting from a repeated multiplication of a factor. Their general form is where is called the base and the exponent.
The base is the factor repeatedly multiplied by itself, and the exponent is the number of times appears in the repeated multiplication.
When a number is written as a power, we say it is written in exponential form.
When a power is written as a repeated multiplication, we say it is written in expanded form.
You should know the product rule and the power of a product rule.
The Product Rule
The product of two powers that have the same base is a power of this same base with an exponent equal to the sum of the exponents:
Power of a Product
A given power of a product of factors is the product of each factor raised to that given power:
Let us start with an example of division of powers of the same base.
Example 1: Quotient of Powers of the Same Base
We notice that, in this expression, all the powers have the same base: 7. By expanding all the powers, we see that the numerator would consist of a repeated multiplication of the factor 7, and the factor 7 would appear times. This can be written as . This is the product rule, which states that . Similarly, we find that the denominator is . We have already simplified our expression to Now, we know that one 7 in the numerator cancels out with one 7 in the denominator since . So, here, the eleven 7s in the numerator cancel out with eleven out of the twelve 7s in the denominator to give 1. This leaves us with
This can be visualized below.
The question now is whether we could express the result of the previous question, , as a power of 7.
So far, we have assumed that the exponent is positive. To understand the meaning of extending the exponents to negative numbers, let us look at what happens when we divide the power of a number by this number, for instance, divided by 7. Given that , we see that If we perform the same division again on , we get
The number is of course simply 7, but it is useful to write the exponent here, as you will see in the following.
So we start seeing a pattern here: every time we divide by 7, the exponent decreases by 1.
If we carry out the division by 7 once more, we get
One further division by 7 leads to and another one leads to
Let us summarize these findings in a table.
We have just found out something very interesting. The exponent, when positive, tells us how many times the base is used in a repeated multiplication of 1. When it is negative, then it tells us how many times the base is used in a repeated division of 1. From this, it follows that any power of a number is the reciprocal of the power with the same base but with an opposite exponent. This is given by the negative exponent rule.
Negative Exponent Rule
Any power can be expressed as the reciprocal (or multiplicative inverse) of a power having the same base but an opposite exponent:
Hence, we find that is a power of 7 with an exponent of : . In the previous example, we have found that and if we remember how pairs of factors in the numerator and denominator cancel out to give 1, we can easily conclude that .
It may be easier to visualize first with this type of example:
Whatever the values of their respective exponents, the quotient of two powers of the same base is given by the quotient rule.
The Quotient Rule
The quotient of two powers that have the same base is a power of this same base with an exponent equal to the difference of the exponents of the dividend and divisor:
Let us practice in the next example how to handle powers with a negative exponent.
Example 2: Evaluating Powers with Negative Exponents
Evaluate the following expression: .
The given expression involves the multiplication of two powers that have a different base, both with a negative exponent.
Recall that any power is the reciprocal of the power having the same base but an opposite exponent. Therefore, and . In other words, a negative exponent tells us how many times the base appears in a repeated division of 1. Hence, and .
We could also have used the converse of the rule for the power of a product, which states that a given power of a product of factors is the product of each factor raised to that given power (which can be written as ). Applied to our expression, we find that
Applying then the negative exponent rule, we find that
Hence, we have found that
In the next example, we are going to see how to deal with negative exponents when applying the product rule.
Example 3: Applying the Product Rule
Simplify , where .
In the given expression, we have a product of powers of the variable . The product rule states that the product of two powers that have the same base is a power of this same base with an exponent equal to the sum of the exponents. Here, we simply need to add the exponents 7, , and 4 to find which power of our expression is equivalent to.
We find that
Let us look now at powers of fractions. For instance, let us consider .
Note that, as the fraction is within parentheses, the base of the power is the whole fraction, in contrast to where only the denominator is a power (it is 3 raised to the fourth power).
We can write in expanded form to get
This expression is a repeated multiplication of , which can be seen also as a repeated multiplication of 3 and a repeated multiplication of or as a repeated multiplication of 3 and a repeated division of 5. Using the commutativity of multiplication, we can rewrite the above expression as or as
Note that we could have found the above expression using the rule for multiplying fractions.
Our last expression can be simplified to or, using our knowledge of negative exponents, to
A power of a fraction is therefore just a special case of a power of a product. Let us generalize our result.
Power of a Fraction
Raising a fraction to a given power is simply given by individually raising its denominator and its numerator:
Let us now look at an example.
Example 4: Evaluating a Product of Powers of a Fraction
Which of the following expressions is equal to ?
The expression given is a product of two powers. We notice that the two powers have the same base, namely, . Applying the product rule of powers, we find that
Now, to evaluate this, we can use the fact that . We need, however, to be careful here because the fraction is negative. We can rewrite it as or as to make sure we do not forget the negative sign. We get then that is equal to or .
Both lead of course to the same result, namely, that
Hence, our answer is that is equivalent to (option E).
The last example is an example of a quotient of powers.
Example 5: Quotient of Powers of Different Bases
Which of the following is equal to ?
We have here an interesting quotient of powers of different bases. Therefore, we cannot apply the quotient rule at first sight. However, looking at the numbers given, we realize that 3 and 2 are factors of 18. So let us replace 18 in the given expression with and see what can be done. We have
Knowing that a given power of a product of factors is the product of each factor raised to that given power, we have
Substituting this in our previous equation, we find
We can now apply the quotient rule; that is,
Applying the negative exponent rule, we finally find that
Hence, the right answer is option E.
- Powers are numbers resulting from a repeated multiplication of a factor. Their general form is , where is called the base and the exponent.
- The product of two powers that have the same base is a power of this same base with an exponent equal to the sum of the exponents:
- A given power of a product of factors is the product of each factor raised to that given power:
- Any power can be expressed as the reciprocal (or multiplicative inverse) of a power having the same base but an opposite exponent:
- Raising a fraction to a given power is simply given by individually raising its denominator and its numerator: