Learning Objective(s)

·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.

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.

Recall that exponents are a way of representing repeated multiplication. For example, the notation 5⁴ can be expanded and written as 5 • 5 • 5 • 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 (2³)(2⁴). Expanding each exponent, this can be rewritten as (2 • 2 • 2) (2 • 2 • 2 • 2) or  2 • 2 • 2 • 2 • 2 • 2 • 2. In exponential form, you would write the product as 2⁷. 

Notice, 7 is the sum of the original two exponents, 3 and 4.

What about (x²)(x⁶)? This can be written as (x • x)(x • x • x • x • x • x) = x • x • x • x • x • x • x • x  or x⁸. And, once again, 8 is the sum of the original two exponents.

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

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

Let's simplify (5²)⁴. In this case, the base is5² and the exponent is 4, so you multiply 5² four times: (5²)⁴  = 5² •5² •5² •5^{2 =} 5⁸ (using the Product Rule - add the exponents).

(5²)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 5⁸. Notice that the new exponent is the same as the product of the original exponents: 2 •4 = 8.

So, (5²)⁴  = 5^2 •4 =  5⁸ (which equals 390,625, if you do the multiplication).

Likewise, (x⁴)³ = x^4 • 3 = x¹².

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, (2³)⁵ = 2¹⁵.

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.

You can rewrite the expression as: . Then you can cancel the common factors of 4 in the numerator and denominator: 

Finally, this expression can be rewritten as 4³ using exponential notation. Notice that the exponent, 3, is the difference between the two exponents in the original expression, 5 and 2.

So,  = 4^5-2 = 4³.

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

or

  = x⁷⁻⁹ = x^-2

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

Notice that  = 4⁰. And we know that   =  = 1. So this may help to explain why 4⁰ = 1.

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

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

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

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.

This reading is taken from the Developmental Math Open Program created by The NROC Project. It is available under a Creative Commons license.