Explainer: Zero and Negative Exponents

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 ๐‘๐‘š, where ๐‘ is called the base and ๐‘š the exponent.

The base ๐‘ is the factor repeatedly multiplied by itself, and the exponent ๐‘š is the number of times ๐‘ 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: ๐‘๐‘šโ‹…๐‘๐‘›=๐‘๐‘š+๐‘›.

Power of a Product

A given power of a product of factors is the product of each factor raised to that given power: (๐‘Ž๐‘)๐‘š=๐‘Ž๐‘šโ‹…๐‘๐‘š.

Let us start with an example of division of powers of the same base.

Example 1: Quotient of Powers of the Same Base

Simplify 73ร—72ร—7673ร—75ร—72ร—72.

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 the factor 7 would appear 3+2+6=11 times. This can be written as 711 . This is the product rule, which states that ๐‘๐‘šโ‹…๐‘๐‘›=๐‘๐‘š+๐‘›. Similarly, we find that the denominator is 712. We have already simplified our expression to 711712. Now, we know that one 7 in the numerator cancels out with one 7 in the denominator since 77=1. 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 171=17.

This can be visualized below. 711712=71โ‹…71โ‹…71โ‹…71โ‹…71โ‹…71โ‹…71โ‹…71โ‹…71โ‹…71โ‹…717โ‹…7โ‹…7โ‹…7โ‹…7โ‹…7โ‹…7โ‹…7โ‹…7โ‹…7โ‹…7โ‹…7=17

The question now is whether we could express the result of the previous question, 17, 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, 73 divided by 7. Given that 73=7โ‹…7โ‹…7, we see that 73รท7=1โ‹…7โ‹…7=72. If we perform the same division again on 72, we get 72รท7=1โ‹…7=71.

The number 71 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 7รท7=1=70.

One further division by 7 leads to 70รท7=1รท7=7โˆ’1 and another one leads to 7โˆ’1รท7=1รท7รท7=7โˆ’2.

Let us summarize these findings in a table.

721โ‹…7โ‹…7
711โ‹…7
701
7โˆ’11รท7=17
7โˆ’21รท7รท7=172

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: ๐‘โˆ’๐‘š=1๐‘๐‘š.

Hence, we find that 17 is a power of 7 with an exponent of โˆ’1: 17=7โˆ’1. In the previous example, we have found that 711712=7โˆ’1, and if we remember how pairs of factors in the numerator and denominator cancel out to give 1, we can easily conclude that 711712=711โˆ’12=7โˆ’1.

It may be easier to visualize first with this type of example: 5652=5โ‹…5โ‹…5โ‹…5โ‹…5โ‹…55โ‹…5=56โˆ’2=54.

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: ๐‘๐‘š๐‘๐‘›=๐‘๐‘šโˆ’๐‘›.

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: 6โˆ’2โ‹…2โˆ’2.

Answer

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, 6โˆ’2=162 and 2โˆ’2=122. In other words, a negative exponent tells us how many times the base appears in a repeated division of 1. Hence, 6โˆ’2=16โ‹…6 and 2โˆ’2=12โ‹…2.

We have 6โˆ’2โ‹…2โˆ’2=16โ‹…6โ‹…12โ‹…2=136โ‹…14=1144.

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 (๐‘Ž๐‘)๐‘š=๐‘Ž๐‘šโ‹…๐‘๐‘š). Applied to our expression, we find that 6โˆ’2โ‹…6โˆ’2=(6โ‹…2)โˆ’2=12โˆ’2.

Applying then the negative exponent rule, we find that 12โˆ’2=1122=1144.

Hence, we have found that 6โˆ’2โ‹…2โˆ’2=1144.

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 ๐‘ฅ7ร—๐‘ฅโˆ’5ร—๐‘ฅ4, where ๐‘ฅโ‰ 0.

Answer

In the given expression, we have a product of powers of the variable ๐‘ฅ. 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, โˆ’5, and 4 to find which power of ๐‘ฅ our expression is equivalent to.

We find that ๐‘ฅ7ร—๐‘ฅโˆ’5ร—๐‘ฅ4=๐‘ฅ7โˆ’5+4=๐‘ฅ6.

Let us look now at powers of fractions. For instance, let us consider ๏€ผ35๏ˆ4.

Note that, as the fraction is within parentheses, the base of the power is the whole fraction, in contrast to 345 where only the denominator is a power (it is 3 raised to the fourth power).

We can write ๏€ผ35๏ˆ4 in expanded form to get 35ร—35ร—35ร—35.

This expression is a repeated multiplication of 35, which can be seen also as a repeated multiplication of 3 and a repeated multiplication of 15 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 3ร—3ร—3ร—3ร—15ร—15ร—15ร—15 or as 3ร—3ร—3ร—35ร—5ร—5ร—5.

Note that we could have found the above expression using the rule for multiplying fractions.

Our last expression can be simplified to 3454, or, using our knowledge of negative exponents, to 34โ‹…5โˆ’4.

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: ๏€ป๐‘Ž๐‘๏‡๐‘š=๐‘Ž๐‘š๐‘๐‘š=๐‘Ž๐‘šโ‹…๐‘โˆ’๐‘š.

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 ๏€ผโˆ’34๏ˆโˆ’3ร—๏€ผโˆ’34๏ˆ6?

  1. ๏€ผโˆ’34๏ˆโˆ’18
  2. 2764
  3. 21027
  4. 7294,096
  5. โˆ’2764

Answer

The expression given is a product of two powers. We notice that the two powers have the same base, namely, ๏€ผโˆ’34๏ˆ. Applying the product rule of powers, we find that ๏€ผโˆ’34๏ˆโˆ’3ร—๏€ผโˆ’34๏ˆ6=๏€ผโˆ’34๏ˆโˆ’3+6=๏€ผโˆ’34๏ˆ3.

Now, to evaluate this, we can use the fact that ๏€ป๐‘Ž๐‘๏‡๐‘š=๐‘Ž๐‘š๐‘๐‘š. We need, however, to be careful here because the fraction is negative. We can rewrite it as (โˆ’1)โ‹…34 or as โˆ’34 to make sure we do not forget the negative sign. We get then that ๏€ผโˆ’34๏ˆ3 is equal to (โˆ’1)3โ‹…3343 or (โˆ’3)343.

Both lead of course to the same result, namely, that ๏€ผโˆ’34๏ˆ3=โˆ’2764.

Hence, our answer is that โˆ’2764 is equivalent to ๏€ผโˆ’34๏ˆโˆ’3ร—๏€ผโˆ’34๏ˆ6 (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 36ร—25188?

  1. 132ร—23
  2. 34ร—24
  3. 132ร—211
  4. 134ร—23
  5. 1310ร—23

Answer

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 2ร—3ร—3 and see what can be done. We have 36ร—25188=36ร—25(2ร—3ร—3)8.

Knowing that a given power of a product of factors is the product of each factor raised to that given power, we have (2ร—3ร—3)8=28ร—38ร—38.

Substituting this in our previous equation, we find 36ร—25188=36ร—2528ร—38ร—38.

We can now apply the quotient rule; that is, 36ร—2528ร—38ร—38=36โˆ’8โˆ’8ร—25โˆ’8=3โˆ’10ร—2โˆ’3.

Applying the negative exponent rule, we finally find that 36ร—25188=1310ร—23.

Hence, the right answer is option E.

Key Points

  1. Powers are numbers resulting from a repeated multiplication of a factor. Their general form is ๐‘๐‘š, where ๐‘ is called the base and ๐‘š the exponent.
  2. 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: ๐‘๐‘šโ‹…๐‘๐‘›=๐‘๐‘š+๐‘›.
  3. A given power of a product of factors is the product of each factor raised to that given power: (๐‘Ž๐‘)๐‘š=๐‘Ž๐‘šโ‹…๐‘๐‘š.
  4. Any power can be expressed as the reciprocal (or multiplicative inverse) of a power having the same base but an opposite exponent: ๐‘โˆ’๐‘š=1๐‘๐‘š.
  5. Raising a fraction to a given power is simply given by individually raising its denominator and its numerator: ๏€ป๐‘Ž๐‘๏‡๐‘š=๐‘Ž๐‘š๐‘๐‘š=๐‘Ž๐‘šโ‹…๐‘โˆ’๐‘š.

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