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.
We should also know the following four rules, or laws, of exponents that we will use in our working.
Rule: Product Rule
The product of two powers with the same base is a power of this same base with an exponent equal to the sum of the exponents:
Rule: Quotient Rule
The quotient of two powers with 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:
Rule: Power of a Product Rule
A given power of a product of factors is the product of each factor raised to that given power:
Rule: Power of a Power Rule
Raising a power to a further power is done by writing the base with an exponent equal to the product of the two exponents:
Let us start with an example of division of powers of the same base.
Example 1: Quotient of Powers of the Same Base
Simplify .
Answer
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 that 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.
| 1 | |
We have just found out some very interesting information, which leads us to two important rules. First, when the exponent is zero, the answer is 1. This result holds true for all nonzero bases.
Rule: Zero Exponent Rule
Any nonzero base with an exponent of zero is equal to 1:
A consequence of this rule is that whenever we see a nonzero number raised to the power of zero, we can always write down its value as 1 without doing any further calculations. In addition, note that by applying the quotient rule in the case , we can derive the zero exponent rule directly: , which simplifies to .
Now, we state the second rule. In the above working, when the exponent is positive, it tells us how many times the base is used in a repeated multiplication of 1. When the exponent is negative, 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 the negative exponent rule.
Rule: Negative Exponent Rule
Any power can be expressed as the reciprocal (or multiplicative inverse) of a power with the same base but opposite exponent:
Hence, we find that is a power of 7 with an exponent of (i.e., ). In the first example, we 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 .
In the next example, let us practice how to handle powers with a negative exponent.
Example 2: Evaluating Powers with Negative Exponents
Evaluate the following expression: .
Answer
The given expression involves the multiplication of two powers that have a different base, both with a negative exponent.
Recall the negative exponent rule, which tells us 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. Thus, and , so we have
Note that as an alternative approach, we could have worked backward from the power of a product rule, which states that a given power of a product of factors is the product of each factor raised to that given power: . Applied to our expression, we find that
Then, applying the negative exponent rule , we find that
Hence, we have found that
Let us look now at powers of fractions. For instance, let us consider .
Note that since the fraction is within parentheses, the base of the power is the whole fraction, in contrast to where only the numerator 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 also be seen 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.
Rule: Power of a Fraction Rule
Raising a fraction to a given power is done by raising its numerator and its denominator individually:
Now, we can try an example.
Example 3: Evaluating Negative Single-Term Rational Expressions with Negative Integer Exponents
Which of the following is equal to ?
Answer
Here, we have a negative fraction with a negative integer exponent and we need to work out its value.
We start by transforming the negative exponent into a positive one. Recall the power of a power rule, which tells us that raising a power to a further power is done by writing the base with an exponent equal to the product of the two exponents:
By writing the exponent as , we can work backward from this rule by taking , , and , giving
Notice that inside the big parentheses, we have a fraction to the power of . Recall the negative exponent rule, which tells us that any power can be expressed as the reciprocal of a power with the same base but opposite exponent: . Applying this to , we see that it is equivalent to . This is the reciprocal of the fraction , which can be found by swapping the numerator and denominator. Thus, we get . So, we have now simplified our original expression to
To evaluate this, we can use the power of a fraction rule, which states that raising a fraction to a given power is done by raising its numerator and its denominator individually:
We need, however, to be careful here because the fraction is negative. Therefore, we can rewrite as or to make sure we do not forget the negative sign. Doing this and then applying the power of a fraction rule, we get
Finally, we evaluate the numerator and denominator separately to get the answer .
We conclude that , so answer E is correct.
So far, we have combined our rules for zero and negative exponents with other rules of exponents to help us evaluate numerical expressions. Very usefully, it turns out that the same rules can also be applied to help us simplify algebraic expressions. Here is an example of this type.
Example 4: Simplifying an Algebraic Expression Involving Negative Powers
Simplify .
Answer
In this question, we are given a quotient comprising two negative powers with the same base, .
Recall the quotient rule, which states that the quotient of two powers with 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:
Here, we have , , and , so applying the rule gives
Therefore, when we simplify , we get .
In our final two questions, the power of a power rule will be especially important in helping us simplify exponents. We start with a numerical example.
Example 5: Evaluating Positive Rational Expressions with Negative Integer Exponents Raised to Positive Integer Exponents
Which of the following expressions has the same value as ?
Answer
The given expression is a negative power of the fraction raised to a further power, which is positive.
Recall the power of a power rule, which tells us that raising a power to a further power is done by writing the base with an exponent equal to the product of the two exponents:
Therefore, we can apply this rule with , , and , which gives
The original expression has now been transformed into a fraction base with a single negative exponent.
It will be helpful if we can find a way to make the exponent positive. Note that we can do this by once more applying the power of a power rule: . By writing the exponent as , we can work backward from this rule by taking , , and , giving
Notice that inside the big parentheses, we have a fraction to the power of . Now, recall the negative exponent rule, which tells us that any power can be expressed as the reciprocal of a power with the same base but opposite exponent: . Applying this to , we see that it is equivalent to . This is the reciprocal of the fraction , which can be found by swapping the numerator and denominator. Thus, we get , which means we have now simplified our original expression to
Finally, we can use the power of a fraction rule, which states that raising a fraction to a given power is done by raising its numerator and its denominator individually:
Here, we have , , and , so
This is now in the right form to compare with the given answer options, so we see that C is correct.
We finish with an example that involves simplifying an algebraic expression by using many of the exponent rules we have explored.
Example 6: Simplifying an Algebraic Expression Involving Negative Powers
Which of the following expressions has the same value as ?
Answer
Here, we have a complicated fraction expression, which we will refer to as the main fraction, raised to the power of . Notice that this main fraction is made up of three powers of the same base, namely the algebraic fraction . Our strategy will be to simplify the main fraction by applying some of the rules of exponents to these three powers.
Starting with the numerator , we see that the first term has a zero exponent. Since the zero exponent rule tells us that any nonzero base with an exponent of zero is equal to 1, then the numerator simplifies as
Alternatively, we could have applied the product rule with , , and to get the same result by a different method: .
Therefore, having simplified the numerator, we can now rewrite the main fraction as
We have thus obtained a quotient of two powers with the same base. Recalling the quotient rule and taking , , and , the main fraction simplifies as follows:
The last stage of simplification is to raise the main fraction to the power of , which gives
Notice that this is now in the form of a power of a power of , as are all the answer options.
Recall that the power of a power rule says that raising a power to a further power is done by writing the base with an exponent equal to the product of the two exponents:
If we apply this rule to our simplified expression by taking , , and , we get
Therefore, the correct answer option will be the one where the product of its two exponents is equal to . Working through them in order, we get
Since the final option gives a result of , we conclude that E is the correct answer.
Let us finish by recapping some key concepts from this explainer.
Key Points
- Powers are numbers resulting from a repeated multiplication of a factor. Their general form is , where is called the base and the exponent.
- By using the laws of exponents, we can derive further ones for zero and negative exponents.
- The zero exponent rule states that any nonzero base with an exponent of zero is equal to 1:
- The negative exponent rule states that any power can be expressed as the reciprocal (or multiplicative inverse) of a power with the same base but opposite exponent:
