11.1.2: Simplify by Using the Product, Quotient, and Power Rules

Learning Objectives

• Use the product rule to multiply exponential expressions with like bases.
• Use the power rule to raise powers to powers.
• Use the quotient rule to divide exponential expressions with like bases.
• Simplify expressions using a combination of the properties.

Introduction

Exponential notation was developed to write repeated multiplication more efficiently. There are times when it is easier to leave the expressions in exponential notation when multiplying or dividing. Let’s look at rules that will allow you to do this.

The Product Rule for Exponents

Recall that exponents are a way of representing repeated multiplication. For example, the notation \(\ 5^{4}\) can be expanded and written as \(\ 5 \cdot 5 \cdot 5 \cdot 5\), or 625. And don’t forget, the exponent only applies to the number immediately to its left, unless there are parentheses.

What happens if you multiply two numbers in exponential form with the same base? Consider the expression \(\ \left(2^{3}\right)\left(2^{4}\right)\). Expanding each exponent, this can be rewritten as \(\ (2 \cdot 2 \cdot 2)(2 \cdot 2 \cdot 2 \cdot 2)\) or \(\ 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2\). In exponential form, you would write the product as \(\ 2^{7}\). Notice, 7 is the sum of the original two exponents, 3 and 4.

What about \(\ \left(x^{2}\right)\left(x^{6}\right)\)? This can be written as \(\ (x \cdot x)(x \cdot x \cdot x \cdot x \cdot x \cdot x)=x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \text { or } x^{8}\). And, once again, 8 is the sum of the original two exponents.

The Product Rule for Exponents

For any number \(\ x\) and any integers \(\ a\) and \(\ b\), \(\ \left(x^{a}\right)\left(x^{b}\right)=x^{a+b}\).

To multiply exponential terms with the same base, simply add the exponents.

Example

Simplify. \(\ \left(a^{3}\right)\left(a^{7}\right)\)

Solution

\(\ \left(a^{3}\right)\left(a^{7}\right)\)	The base of both exponents is \(\ a\), so the product rule applies.
\(\ a^{3+7}\)	Add the exponents with a common base.

\(\ \left(a^{3}\right)\left(a^{7}\right)=a^{10}\)

When multiplying more complicated terms, multiply the coefficients and then multiply the variables.

Example

Simplify. \(\ 5 a^{4} \cdot 7 a^{6}\)

Solution

\(\ 35 \cdot a^{4} \cdot a^{6}\)	Multiply the coefficients.
\(\ 35 \cdot a^{4+6}\)	The base of both exponents is \(\ a\), so the product rule applies. Add the exponents.
\(\ 35 \cdot a^{10}\)	Add the exponents with a common base.

\(\ 5 a^{4} \cdot 7 a^{6}=35 a^{10}\)

Exercise

Simplify the expression, keeping the answer in exponential notation.

\(\ \left(4 x^{5}\right)\left(2 x^{8}\right)\)

1. \(\ 8 x^{5} \cdot x^{8}\)
2. \(\ 6 x^{13}\)
3. \(\ 8 x^{13}\)
4. \(\ 8 x^{40}\)

Answer

1. Incorrect. \(\ 8 x^{5} \cdot x^{8}\) is equivalent to \(\ \left(4 x^{5}\right)\left(2 x^{8}\right)\), but it still is not in simplest form. Simplify \(\ x^{5} \cdot x^{8}\) by using the Product Rule to add exponents. The correct answer is \(\ 8 x^{13}\).
2. Incorrect. \(\ 6 x^{13}\) is not equivalent to \(\ \left(4 x^{5}\right)\left(2 x^{8}\right)\). In this incorrect response, the correct exponents were added, but the coefficients were also added together. They should have been multiplied. The correct answer is \(\ 8 x^{13}\).
3. Correct. \(\ 8 x^{13}\) is equivalent to \(\ \left(4 x^{5}\right)\left(2 x^{8}\right)\). Multiply the coefficients \(\ (4 \cdot 2)\) and apply the Product Rule to add the exponents of the variables (in this case \(\ x\)) that are the same.
4. Incorrect. \(\ 8 x^{40}\) is not equivalent to \(\ \left(4 x^{5}\right)\left(2 x^{8}\right)\). Do not multiply the coefficients and the exponents. Remember, using the Product Rule add the exponents when the bases are the same. The correct answer is \(\ 8 x^{13}\).

The Power Rule for Exponents

Let’s simplify \(\ \left(5^{2}\right)^{4}\). In this case, the base is \(\ 5^{2}\) and the exponent is 4, so you multiply \(\ 5^{2}\) four times: \(\ \left(5^{2}\right)^{4}=5^{2} \cdot 5^{2} \cdot 5^{2} \cdot 5^{2}=5^{8}\) (Using the Product Rule, add the exponents).

\(\ \left(5^{2}\right)^{4}\) is a power of a power. It is the fourth power of 5 to the second power. And we saw above that the answer is \(\ 5^{8}\). Notice that the new exponent is the same as the product of the original exponents: \(\ 2 \cdot 4=8\).

So, \(\ \left(5^{2}\right)^{4}=5^{2 \cdot 4}=5^{8}\) (which equals 390, 625, if you do the multiplication).

Likewise, \(\ \left(x^{4}\right)^{3}=x^{4 \cdot 3}=x^{12}\).

This leads to another rule for exponents: the Power Rule for Exponents. To simplify a power of a power, you multiply the exponents, keeping the base the same. For example, \(\ \left(2^{3}\right)^{5}=2^{15}\).

The Power Rule for Exponents

For any positive number \(\ x\) and integers \(\ a\) and \(\ b\):

\(\ \left(x^{a}\right)^{b}=x^{a \cdot b}\)

Example

Simplify. \(\ 6\left(c^{4}\right)^{2}\)

Solution

\(\ 6\left(c^{4}\right)^{2}\)

Since you are raising a power to a power, apply the Power Rule and multiply exponents to simplify. The coefficient remains unchanged because it is outside of the parentheses.

\(\ 6\left(c^{4}\right)^{2}=6 c^{8}\)

Example

Simplify. \(\ a^{2}\left(a^{5}\right)^{3}\)

Solution

\(\ a^{2} a^{5 \cdot 3}\)	Raise \(\ a^{5}\) to the power of 3 by multiplying the exponents together (the Power Rule).
\(\ \begin{array}{l} a^{2} a^{15} \\ a^{2+15} \end{array}\)	Since the exponents share the same base, \(\ a\), they can be combined (the Product Rule).

\(\ a^{2}\left(a^{5}\right)^{3}=a^{17}\)

Exercise

Simplify: \(\ -a\left(a^{2}\right)^{4}\)

1. \(\ -a \cdot a^{8}\)
2. \(\ a^{7}\)
3. \(\ -a^{7}\)
4. \(\ -a^{9}\)

Answer

1. Incorrect. This expression is not simplified yet. Recall that \(\ -a\) can also be written \(\ -a^{1}\). Multiply \(\ -a^{1}\) by \(\ a^{8}\) to arrive at the correct answer. The correct answer is \(\ -a^{9}\).
2. Incorrect. Do not add the exponents of 2 and 4 together. The Power Rule states that for a power of a power, you multiply the exponents. The correct answer is \(\ -a^{9}\).
3. Incorrect. Do not add the exponents of 2 and 4 together. The Power Rule states that for a power of a power, you multiply the exponents. The correct answer is \(\ -a^{9}\).
4. Correct. Using the Power Rule, \(\ -a\left(a^{2}\right)^{4}=-a \cdot a^{2 \cdot 4}=-a^{1} a^{8}=-a^{1+8}=-a^{9}\).

The Quotient Rule for Exponents

Let’s look at dividing terms containing exponential expressions. What happens if you divide two numbers in exponential form with the same base? Consider the following expression.

\(\ \frac{4^{5}}{4^{2}}\)

You can rewrite the expression as: \(\ \frac{4 \cdot 4 \cdot 4 \cdot 4 \cdot 4}{4 \cdot 4}\). Then you can cancel the common factors of 4 in the numerator and denominator: \(\ \frac{\not4\cdot\not4\cdot4\cdot4\cdot4}{\not4\cdot\not4}=\frac{4\cdot4\cdot4}{1}\).

Finally, this expression can be rewritten as \(\ 4^{3}\) using exponential notation. Notice that the exponent, 3, is the difference between the two exponents in the original expression, 5 and 2.

So, \(\ \frac{4^{5}}{4^{2}}=4^{5-2}=4^{3}\).

Be careful that you subtract the exponent in the denominator from the exponent in the numerator.

\(\ \frac{x^7}{x^9}=\frac{\not x\cdot\not x\cdot\not x\cdot\not x\cdot\not x\cdot\not x\cdot\not x}{\not x\cdot\not x\cdot\not x\cdot\not x\cdot\not x\cdot\not x\cdot\not x\cdot x\cdot x}=\frac{1}{x\cdot x}=\frac{1}{x^2}=x^{-2}\)

or

\(\ \frac{x^{7}}{x^{9}}=x^{7-9}=x^{-2}\)

So, to divide two exponential terms with the same base, subtract the exponents.

The Quotient Rule for Exponents

For any non-zero number \(\ x\) and any integers \(\ a\) and \(\ b\): \(\ \frac{x^{a}}{x^{b}}=x^{a-b}\)

Notice that \(\ \frac{4^{2}}{4^{2}}=4^{0}\). And we know that \(\ \frac{4^{2}}{4^{2}}=\frac{16}{16}=1\). So this may help to explain why \(\ 4^{0}=1\).

Example

Evaluate. \(\ \frac{4^{9}}{4^{4}}\)

Solution

\(\ 4^{9-4}\)

These two exponents have the same base, 4. According to the Quotient Rule, you can subtract the power in the denominator from the power in the numerator.

\(\ \frac{4^{9}}{4^{4}}=4^{5}\)

When dividing terms that also contain coefficients, divide the coefficients and then divide variable powers with the same base by subtracting the exponents.

Example

Simplify. \(\ \frac{12 x^{4}}{2 x}\)

Solution

\(\ \left(\frac{12}{2}\right)\left(\frac{x^{4}}{x}\right)\)	Separate into numerical and variable factors.
\(\ 6 \cdot x^{4-1}\)	Since the bases of the exponents are the same, you can apply the Quotient Rule. Divide the coefficients and subtract the exponents of matching variables.

\(\ \frac{12 x^{4}}{2^{x}}=6 x^{3}\)

Applying the Rules

All of these rules of exponents, the Product Rule, the Power Rule, and the Quotient Rule, are helpful when evaluating expressions with common bases.

Example

Evaluate \(\ \frac{24 x^{8}}{2 x^{5}}\) when \(\ x=4\).

Solution

\(\ \left(\frac{24}{2}\right)\left(\frac{x^{8}}{x^{5}}\right)\)	Separate into numerical and variable factors.
\(\ 12 \cdot x^{8-5}\)	Divide coefficients, and subtract the exponents of the variables.
\(\ 12 x^{3}\)	Simplify.
\(\ \text { (12) }\left(4^{3}\right)=12 \cdot 64\)	Substitute the value 4 for the variable \(\ x\).

\(\ \frac{24 x^{8}}{2 x^{5}}=768\)

Usually, it is easier to simplify the expression before substituting any values for your variables, but you will get the same answer either way.

Example

Simplify. \(\ \frac{a^{2}\left(a^{5}\right)^{3}}{8 a^{8}}\)

Solution

\(\ \frac{a^{2} \cdot a^{5 \cdot 3}}{8 a^{8}}\)	Use the order of operations with PEMDAS: E: Evaluate exponents. Use the Power Rule to simplify \(\ \left(a^{5}\right)^{3}\)
\(\ \begin{array}{c} \frac{a^{2} a^{15}}{8 a^{8}} \\ \frac{a^{2+15}}{8 a^{8}} \\ \frac{a^{17}}{8 a^{8}} \\ \frac{a^{17-8}}{8} \end{array}\)	M: Multiply, using the Product Rule as the bases are the same. D: Divide using the Quotient Rule.

\(\ \frac{a^{2}\left(a^{5}\right)^{3}}{8 a^{8}}=\frac{a^{9}}{8}\)

Summary

There are rules that help when multiplying and dividing exponential expressions with the same base. To multiply two exponential terms with the same base, add their exponents. To raise a power to a power, multiply the exponents. To divide two exponential terms with the same base, subtract the exponents.