### Learning Objectives

- Use the product rule to multiply exponential expressions
- Use the quotient rule to divide exponential expressions

## Use the product rule to multiply exponential expressions

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

For example, the notation [latex]5^{4}[/latex] can be expanded and written as [latex]5\cdot5\cdot5\cdot5[/latex], 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 [latex]{2}^{3}{2}^{4}[/latex]. Expanding each exponent, this can be rewritten as [latex]\left(2\cdot2\cdot2\right)\left(2\cdot2\cdot2\cdot2\right)[/latex] or [latex]2\cdot2\cdot2\cdot2\cdot2\cdot2\cdot2[/latex]. In exponential form, you would write the product as [latex]2^{7}[/latex]. Notice that 7 is the sum of the original two exponents, 3 and 4.

What about [latex]{x}^{2}{x}^{6}[/latex]? This can be written as [latex]\left(x\cdot{x}\right)\left(x\cdot{x}\cdot{x}\cdot{x}\cdot{x}\cdot{x}\right)=x\cdot{x}\cdot{x}\cdot{x}\cdot{x}\cdot{x}\cdot{x}\cdot{x}[/latex] or [latex]x^{8}[/latex]. And, once again, 8 is the sum of the original two exponents. This concept can be generalized in the following way:

### The Product Rule for Exponents

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

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

Caution! When you are reading mathematical rules, it is important to pay attention to the conditions on the rule. For example, when using the product rule, you may only apply it when the terms being multiplied have the same base and the exponents are integers. Conditions on mathematical rules are often given before the rule is stated, as in this example it says “For any number *x*, and any integers *a* and *b*.”

### Example

Simplify.

[latex](a^{3})(a^{7})[/latex]

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

### Example

Simplify.

[latex]5a^{4}\cdot7a^{6}[/latex]

Caution! Do not try to apply this rule to sums.

Think about the expression [latex]\left(2+3\right)^{2}[/latex]

Does [latex]\left(2+3\right)^{2}[/latex] equal [latex]2^{2}+3^{2}[/latex]?

No, it does not because of the order of operations!

[latex]\left(2+3\right)^{2}=5^{2}=25[/latex]

and

[latex]2^{2}+3^{2}=4+9=13[/latex]

Therefore, you can only use this rule when the numbers inside the parentheses are being multiplied (or divided, as we will see next).

## Use the quotient rule to divide exponential expressions

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.

[latex] \displaystyle \frac{{{4}^{5}}}{{{4}^{2}}}[/latex]

You can rewrite the expression as: [latex] \displaystyle \frac{4\cdot 4\cdot 4\cdot 4\cdot 4}{4\cdot 4}[/latex]. Then you can cancel the common factors of 4 in the numerator and denominator: [latex] \displaystyle [/latex]

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

So, [latex] \displaystyle \frac{{{4}^{5}}}{{{4}^{2}}}=4^{5-2}=4^{3}[/latex].

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

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

### The Quotient (Division) Rule for Exponents

For any non-zero number *x* and any integers *a* and *b*: [latex] \displaystyle \frac{{{x}^{a}}}{{{x}^{b}}}={{x}^{a-b}}[/latex]

### Example

Evaluate. [latex] \displaystyle \frac{{{4}^{9}}}{{{4}^{4}}}[/latex]

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. [latex] \displaystyle \frac{12{{x}^{4}}}{2x}[/latex]

In the following video we show another example of how to use the quotient rule to divide exponential expressions