ORCCA Exponent Rules

Section 6.2 Exponent Rules

Subsection 6.2.1 Review of Exponent Rules for Products and Exponents

In Section 6.1, we introduced three basic rules involving products and exponents. We'll begin with a brief recap and explanation of these three exponent rules.

List 6.2.1. Summary of Exponent Rules

Product Rule: When multiplying two expressions that have the same base, simplify the product by adding the exponents.

\begin{equation*} x^m \cdot x^n = x^{m+n} \end{equation*}
Power to a Power Rule: When a base is raised to an exponent and that expression is raised to another exponent, multiply the exponents.

\begin{equation*} \left(x^m\right)^n = x^{m \cdot n} \end{equation*}
Product to a Power Rule: When a product is raised to an exponent, apply the exponent to each factor in the product.

\begin{equation*} \left(x\cdot y\right)^n = x^{n}\cdot y^{n} \end{equation*}

Checkpoint 6.2.2.

Subsection 6.2.2 Quotients and Exponents

Since division is a form of multiplication, it should seem natural that there are some exponent rules for division as well. Not only are there division rules, these rules for division and exponents are direct counterparts for some of the product rules for exponents.

Quotient of Powers.

When we multiply the same base raised to powers, we end up adding the exponents, as in \(2^{2}\cdot2^{3}=2^{5}\) since \(4\cdot8=32\text{.}\) What happens when we divide the same base raised to powers?

Example 6.2.3.

Simplify \(\frac{x^5}{x^2}\) by first writing out what each power means.

Explanation

Without knowing a rule for simplifying this quotient of powers, we can write the expressions without exponents and simplify.

\begin{align*} \frac{x^5}{x^2} \amp= \frac{x \cdot x \cdot x \cdot x \cdot x}{x \cdot x}\\ \amp= \frac{\cancel{x} \cdot \cancel{x} \cdot x \cdot x \cdot x}{\cancel{x} \cdot \cancel{x} \cdot 1}\\ \amp= \frac{x \cdot x \cdot x}{1}\\ \amp= x^3 \end{align*}

Notice that the difference of the exponents of the numerator and the denominator (\(5\) and \(2\text{,}\) respectively) is \(3\text{,}\) which is the exponent of the simplified expression.

When we divide as we've just done, we end up canceling factors from the numerator and denominator one-for-one. These common factors cancel to give us factors of \(1\text{.}\) The general rule for this is:

Fact 6.2.4. Quotient of Powers Rule.

For any non-zero real number \(a\) and integers \(m\) and \(n\text{,}\)

\begin{equation*} \frac{a^{n}}{a^{m}} = a^{n-m} \end{equation*}

This rule says that when you're dividing two expressions that have the same base, you can simplify the quotient by subtracting the exponents. In Example 6.2.3, this means that we can directly compute \(\frac{x^5}{x^2}\text{:}\)

\begin{align*} \frac{x^5}{x^2} \amp= x^{5-2}\\ \amp=x^3 \end{align*}

Quotient to a Power.

Another rule we have learned is the product to a power rule, which applies the outer exponent to each factor in the product inside the parentheses. We can use the rules of fractions to extend this property to a quotient raised to a power.

Example 6.2.5.

Let \(y\) be a real number, where \(y \neq 0\text{.}\) Find another way to write \(\left(\frac{7}{y}\right)^4\text{.}\)

Explanation

Writing the expression without an exponent and then simplifying, we have:

\begin{align*} \left( \frac{7}{y} \right)^4 \amp= \left( \frac{7}{y} \right) \left( \frac{7}{y} \right) \left( \frac{7}{y} \right) \left( \frac{7}{y} \right)\\ \amp= \frac{7 \cdot 7 \cdot 7 \cdot 7}{y \cdot y \cdot y \cdot y}\\ \amp= \frac{7^4}{y^4}\\ \amp= \frac{2401}{y^4} \end{align*}

Similar to the product to a power rule, we essentially applied the outer exponent to the “factors” inside the parentheses—to factors of the numerator and factors of the denominator. The general rule is:

Fact 6.2.6. Quotient to a Power Rule.

For real numbers \(a\) and \(b\) (with \(b \neq 0\)) and integer \(n\text{,}\)

\begin{equation*} \left( \frac{a}{b} \right)^{n} = \frac{a^{n}}{b^{n}} \end{equation*}

This rule says that when you raise a fraction to a power, you may separately raise the numerator and denominator to that power. In Example 6.2.5, this means that we can directly calculate \(\left( \frac{7}{y} \right)^4\text{:}\)

\begin{align*} \left( \frac{7}{y} \right)^4 \amp= \frac{7^4}{y^4}\\ \amp=\frac{2401}{y^4} \end{align*}

Practice.

Try these exercises that use the quotient rules for exponents.

Checkpoint 6.2.7.

Subsection 6.2.3 The Zero Exponent

So far, we have been working with exponents that are natural numbers (\(1, 2, 3, \ldots\)). By the end of this chapter, we will expand our understanding to include exponents that are any integer, including \(0\) and negative numbers. As a first step, we will focus on understanding how \(0\) should behave as an exponent by considering the pattern of decreasing powers of \(2\) below.

Table 6.2.8. Descending Powers of \(2\)

power		product		value
\(2^4\)	\(=\)	\(2 \cdot 2 \cdot 2 \cdot 2\)	\(=\)	\(16\)	(divide by \(2\))
\(2^3\)	\(=\)	\(2 \cdot 2 \cdot 2\)	\(=\)	\(8\)	(divide by \(2\))
\(2^2\)	\(=\)	\(2 \cdot 2\)	\(=\)	\(4\)	(divide by \(2\))
\(2^1\)	\(=\)	\(2\)	\(=\)	\(2\)	(divide by \(2\))
\(2^0\)	\(=\)	\(\mathord{?}\)	\(=\)	\(\mathord{?}\)

As we move down from one row to the row below it, we reduce the power by \(1\) and we remove a factor of \(2\text{.}\) The question then becomes, “What happens when you remove the only remaining factor of \(2\text{,}\) when you have no factors of 2?” We can see that “removing a factor of \(2\)” really means that we're dividing the value by \(2\text{.}\) Following that pattern, we can see that moving from \(2^1\) to \(2^0\) means that we need to divide the value \(2\) by \(2\text{.}\) Since \(2\div 2 = 1\text{,}\) we have:

\begin{equation*} 2^0 = 1 \end{equation*}

Fact 6.2.9. The Zero Exponent Rule.

For any non-zero real number \(a\text{,}\)

\begin{equation*} a^0 = 1 \end{equation*}

We exclude the case where \(a=0\) from this rule, because our reasoning for this rule with the table had us dividing by the base. And we cannot divide by \(0\text{.}\)

Checkpoint 6.2.10.

Subsection 6.2.4 Negative Exponents

In Section 6.1, we developed rules for simplifying expressions with whole number exponents, like \(0\text{,}\) \(1\text{,}\) \(2\text{,}\) \(3\text{,}\) etc. It turns out that these same rules apply even if the exponent is a negative integer, like \(-1\text{,}\) \(-2\text{,}\) \(-3\text{,}\) etc.

To consider the effects of negative integer exponents, let's extend the pattern we examined in Table 6.2.8. In that table, each time we move down a row, we reduce the power by \(1\) and we divide the value by \(2\text{.}\) We can continue this pattern in the power and value columns, paying particular attention to the values for negative exponents:

Power	Value
\(2^3\)	\(8\)	(divide by \(2\))
\(2^2\)	\(4\)	(divide by \(2\))
\(2^1\)	\(2\)	(divide by \(2\))
\(2^0\)	\(1\)	(divide by \(2\))
\(2^{-1}\)	\(\sfrac{1}{2}=\sfrac{1}{2^1}\)	(divide by \(2\))
\(2^{-2}\)	\(\sfrac{1}{4}=\sfrac{1}{2^2}\)	(divide by \(2\))
\(2^{-3}\)	\(\sfrac{1}{8}=\sfrac{1}{2^3}\)

Note that the choice of base \(2\) was arbitrary, and this pattern works for all bases except \(0\text{,}\) since we cannot divide by \(0\) in moving from one row to the next.

Fact 6.2.11. The Negative Exponent Rule.

For any non-zero real number \(a\) and any integer \(n\text{,}\)

\begin{equation*} a^{-n} = \frac{1}{a^n} \end{equation*}

Note that if we take reciprocals of both sides, we have another helpful fact:

\begin{equation*} \frac{1}{a^{-n}}=a^n\text{.} \end{equation*}

Taken together, these facts tell us that a negative exponent power in the numerator belongs in the denominator (with a positive exponent) and a negative exponent power in the denominator belongs in the numerator (with a positive exponent). In other words, you can see a negative exponent as telling you to move things in and out of the numerator and denominator of an expression.

Remark 6.2.12.

You may be expected to simplify expressions so that they do not have any negative exponents. This can always be accomplished using the negative exponent rule.

Try these exercises that involve negative exponents.

Checkpoint 6.2.13.

Subsection 6.2.5 Summary of Exponent Rules

Now that we have some new exponent rules beyond those from Section 6.1, let's summarize.

List 6.2.14. Summary of the Rules of Exponents for Multiplication and Division

If \(a\) and \(b\) are real numbers, and \(n\) and \(m\) are integers, then we have the following rules:

Product Rule: \(\displaystyle a^{n} \cdot a^{m} = a^{n+m}\)
Power to a Power Rule: \(\displaystyle (a^{n})^{m} = a^{n\cdot m}\)
Product to a Power Rule: \(\displaystyle (ab)^{n} = a^{n} \cdot b^{n}\)
Quotient Rule: \(\dfrac{a^{n}}{a^{m}} = a^{n-m}\text{,}\) as long as \(a \neq 0\)
Quotient to a Power Rule: \(\left( \dfrac{a}{b} \right)^{n} = \dfrac{a^{n}}{b^{n}}\text{,}\) as long as \(b \neq 0\)
Zero Exponent Rule: \(a^{0} = 1\) for \(a\neq0\)
Negative Exponent Rule: \(\displaystyle a^{-n} = \frac{1}{a^n}\)
Negative Exponent Reciprocal Rule: \(\displaystyle \frac{1}{a^{-n}} = a^n\)

Remark 6.2.15. Why we have “\(a \neq 0\)” and “\(b \neq 0\)” for some rules.

Whenever we're working with division, we have to be careful to make sure the rules we state don't ever imply that we might be dividing by zero. Dividing by zero leads us to expressions that have no meaning. For example, both \(\frac{9}{0}\) and \(\frac{0}{0}\) are undefined, meaning no one has defined what it means to divide a number by \(0\text{.}\) Also, we established that \(a^0=1\) using repeated division by \(a\) in table rows, so that reasoning doesn't work if \(a=0\text{.}\)

Warning 6.2.16. A Common Mistake.

It may be tempting to apply the rules of exponents to expressions containing addition or subtraction. However, none of the rules of exponents involve addition or subtraction in the initial expression. Because whole number exponents mean repeated multiplication, not repeated addition or subtraction, trying to apply exponent rules in situations that do not use multiplication simply doesn't work.

Can we say something like \(a^n + a^m = a^{n+m}\text{?}\) How would that work out when \(a=2\text{?}\)

\begin{align*} 2^3 + 2^4 \amp\stackrel{?}{=} 2^{3+4}\\ 8 + 16 \amp\stackrel{?}{=} 2^{7}\\ 24 \amp\neq 128 \end{align*}

As we can see, that's not even close. This attempt at a “sum rule” falls apart. In fact, without knowing values for \(a\text{,}\) \(n\text{,}\) and \(m\text{,}\) there's no way to simplify the expression \(a^n + a^m\text{.}\)

Checkpoint 6.2.17.

As we mentioned before, many situations we'll come across will require us to use more than one exponent rule. In these situations, we'll have to decide which rule to use first. There are often different, correct approaches we could take. But if we rely on order of operations, we will have a straightforward approach to simplify the expression correctly. To bring it all together, try these exercises.

Checkpoint 6.2.18.

Exercises 6.2.6 Exercises

Review and Warmup

1.

Evaluate the following.

\(3^{2}\)
\(5^{3}\)
\((-5)^{2}\)
\((-2)^{3}\)

2.

Evaluate the following.

\(4^{2}\)
\(3^{3}\)
\((-4)^{2}\)
\((-3)^{3}\)

3.

Use the properties of exponents to simplify the expression.

\({7}\cdot{7^{2}}\)

4.

Use the properties of exponents to simplify the expression.

\({7}\cdot{7^{8}}\)

5.

Use the properties of exponents to simplify the expression.

\({8^{5}}\cdot{8^{8}}\)

6.

Use the properties of exponents to simplify the expression.

\({9^{2}}\cdot{9^{3}}\)

Simplifying Products and Quotients Involving Exponents

7.

Use the properties of exponents to simplify the expression.

\({r^{3}}\cdot{r^{14}}\)

8.

Use the properties of exponents to simplify the expression.

\({t^{5}}\cdot{t^{8}}\)

9.

Use the properties of exponents to simplify the expression.

\(\displaystyle{\left(y^{5}\right)^{12}}\)

10.

Use the properties of exponents to simplify the expression.

\(\displaystyle{\left(t^{6}\right)^{9}}\)

11.

Use the properties of exponents to simplify the expression.

\(\left(2x^{7}\right)^2\)

12.

Use the properties of exponents to simplify the expression.

\(\left(4t^{8}\right)^4\)

13.

Use the properties of exponents to simplify the expression.

\(\displaystyle{({3r^{15}})\cdot({6r^{13}})}\)

14.

Use the properties of exponents to simplify the expression.

\(\displaystyle{({-5r^{18}})\cdot({8r^{6}})}\)

15.

Use the properties of exponents to simplify the expression.

\(\displaystyle{\left({-\frac{t^{20}}{9}}\right) \cdot \left({\frac{t^{19}}{6}}\right)}\)

16.

Use the properties of exponents to simplify the expression.

\(\displaystyle{\left({\frac{t^{3}}{5}}\right) \cdot \left({-\frac{t^{12}}{5}}\right)}\)

17.

Use the properties of exponents to simplify the expression.

\(-3\left(-4y^{3}\right)^2\)

18.

Use the properties of exponents to simplify the expression.

\(-4\left(-9y^{5}\right)^3\)

19.

Use the properties of exponents to simplify the expression.

\(\left(-31\right)^0=\)

20.

Use the properties of exponents to simplify the expression.

\(\left(-25\right)^0=\)

21.

Use the properties of exponents to simplify the expression.

\(-32^0=\)

22.

Use the properties of exponents to simplify the expression.

\(-37^0=\)

23.

Use the properties of exponents to simplify the expression.

\(43^0+\left(-43\right)^0=\)

24.

Use the properties of exponents to simplify the expression.

\(48^0+\left(-48\right)^0=\)

25.

Use the properties of exponents to simplify the expression.

\(5b^0=\)

26.

Use the properties of exponents to simplify the expression.

\(10y^0=\)

27.

Use the properties of exponents to simplify the expression.

\(\left(-428m\right)^0=\)

28.

Use the properties of exponents to simplify the expression.

\(\left(-207b\right)^0=\)

29.

Use the properties of exponents to simplify the expression.

\(\left(\displaystyle\frac{x^{2}}{6}\right)^{3}=\)

30.

Use the properties of exponents to simplify the expression.

\(\left(\displaystyle\frac{x^{8}}{7}\right)^{2}=\)

31.

Use the properties of exponents to simplify the expression.

\(\left(\displaystyle\frac{-7}{4x^{7}}\right)^{3}=\)

32.

Use the properties of exponents to simplify the expression.

\(\left(\displaystyle\frac{-9}{10x^{10}}\right)^{2}=\)

33.

Use the properties of exponents to simplify the expression.

\(\left(\displaystyle\frac{5x^{10}}{2}\right)^{3}=\)

34.

Use the properties of exponents to simplify the expression.

\(\left(\displaystyle\frac{9x^{2}}{10}\right)^{2}=\)

35.

Use the properties of exponents to simplify the expression.

\(\displaystyle\frac{{-24r^{12}}}{{4r^{3}}}=\)

36.

Use the properties of exponents to simplify the expression.

\(\displaystyle\frac{{20x^{7}}}{{5x^{2}}}=\)

37.

Use the properties of exponents to simplify the expression.

\(\displaystyle\frac{{7x^{4}}}{{21x^{3}}}=\)

38.

Use the properties of exponents to simplify the expression.

\(\displaystyle\frac{{9x^{11}}}{{54x^{8}}}=\)

39.

Use the properties of exponents to simplify the expression.

\(\displaystyle\frac{{t^{14}}}{{t^{8}}}=\)

40.

Use the properties of exponents to simplify the expression.

\(\displaystyle\frac{{y^{16}}}{{y^{4}}}=\)

41.

Use the properties of exponents to simplify the expression.

\(\displaystyle\frac{{19^{19}}}{{19^{13}}}=\)

42.

Use the properties of exponents to simplify the expression.

\(\displaystyle\frac{{20^{15}}}{{20^{3}}}=\)

43.

Use the properties of exponents to simplify the expression.

\(\left(\displaystyle\frac{x^{3}}{2y^{7}z^{4}}\right)^{2}=\)

44.

Use the properties of exponents to simplify the expression.

\(\left(\displaystyle\frac{x^{9}}{2y^{10}z^{2}}\right)^{2}=\)

45.

Use the properties of exponents to simplify the expression.

\(\left(\displaystyle\frac{-5x^{4}}{2y^{6}}\right)^{2}=\)

46.

Use the properties of exponents to simplify the expression.

\(\left(\displaystyle\frac{-3x^{5}}{10y^{3}}\right)^{3}=\)

47.

Use the properties of exponents to simplify the expression.

\(\displaystyle\frac{{15^{17}}\cdot{11^{5}}}{{15^{5}}\cdot{11^{2}}}=\)

48.

Use the properties of exponents to simplify the expression.

\(\displaystyle\frac{{16^{11}}\cdot{20^{5}}}{{16^{7}}\cdot{20^{3}}}=\)

49.

Use the properties of exponents to simplify the expression.

\(\displaystyle\frac{{-30x^{16}y^{12}z^{17}}}{{15x^{5}y^{10}z^{5}}}=\)

50.

Use the properties of exponents to simplify the expression.

\(\displaystyle\frac{{-85x^{13}y^{20}z^{7}}}{{17x^{5}y^{18}z^{4}}}=\)

51.

Use the properties of exponents to simplify the expression.

\(\displaystyle\frac{{57x^{5}y^{11}}}{{19x^{4}y}}=\)