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Study Guides > College Algebra

Rules for Exponents

Learning Objectives

  • Use the product rule for exponents
  • Use the quotient rule for exponents
  • Use the power rule for exponents
Consider the product x3⋅x4{x}^{3}\cdot {x}^{4}. Both terms have the same base, x, but they are raised to different exponents. Expand each expression, and then rewrite the resulting expression.
\begin{array}\text{ }x^{3}\cdot x^{4}\hfill&=\stackrel{\text{3 factors } \text{ 4 factors}}{x\cdot x\cdot x\cdot x\cdot x\cdot x\cdot x} \\ \hfill& =\stackrel{7 \text{ factors}}{x\cdot x\cdot x\cdot x\cdot x\cdot x\cdot x} \\ \hfill& =x^{7}\end{array}
The result is that x3⋅x4=x3+4=x7{x}^{3}\cdot {x}^{4}={x}^{3+4}={x}^{7}. Notice that the exponent of the product is the sum of the exponents of the terms. In other words, when multiplying exponential expressions with the same base, we write the result with the common base and add the exponents. This is the product rule of exponents.
am⋅an=am+n{a}^{m}\cdot {a}^{n}={a}^{m+n}
Now consider an example with real numbers.
23⋅24=23+4=27{2}^{3}\cdot {2}^{4}={2}^{3+4}={2}^{7}
We can always check that this is true by simplifying each exponential expression. We find that 23{2}^{3} is 8, 24{2}^{4} is 16, and 27{2}^{7} is 128. The product 8⋅168\cdot 16 equals 128, so the relationship is true. We can use the product rule of exponents to simplify expressions that are a product of two numbers or expressions with the same base but different exponents.

A General Note: The Product Rule of Exponents

For any real number aa and natural numbers mm and nn, the product rule of exponents states that
am⋅an=am+n{a}^{m}\cdot {a}^{n}={a}^{m+n}

Example: Using the Product Rule

Write each of the following products with a single base. Do not simplify further.
  1. t5⋅t3{t}^{5}\cdot {t}^{3}
  2. (−3)5⋅(−3)\left(-3\right)^{5}\cdot \left(-3\right)
  3. x2⋅x5⋅x3{x}^{2}\cdot {x}^{5}\cdot {x}^{3}

Answer: Use the product rule to simplify each expression.

  1. t5⋅t3=t5+3=t8{t}^{5}\cdot {t}^{3}={t}^{5+3}={t}^{8}
  2. (−3)5⋅(−3)=(−3)5⋅(−3)1=(−3)5+1=(−3)6{\left(-3\right)}^{5}\cdot \left(-3\right)={\left(-3\right)}^{5}\cdot {\left(-3\right)}^{1}={\left(-3\right)}^{5+1}={\left(-3\right)}^{6}
  3. x2⋅x5⋅x3{x}^{2}\cdot {x}^{5}\cdot {x}^{3}
At first, it may appear that we cannot simplify a product of three factors. However, using the associative property of multiplication, begin by simplifying the first two.
x2⋅x5⋅x3=(x2⋅x5)⋅x3=(x2+5)⋅x3=x7⋅x3=x7+3=x10{x}^{2}\cdot {x}^{5}\cdot {x}^{3}=\left({x}^{2}\cdot {x}^{5}\right)\cdot {x}^{3}=\left({x}^{2+5}\right)\cdot {x}^{3}={x}^{7}\cdot {x}^{3}={x}^{7+3}={x}^{10}
Notice we get the same result by adding the three exponents in one step.
x2⋅x5⋅x3=x2+5+3=x10{x}^{2}\cdot {x}^{5}\cdot {x}^{3}={x}^{2+5+3}={x}^{10}

Try It

Write each of the following products with a single base. Do not simplify further.
  1. k6⋅k9{k}^{6}\cdot {k}^{9}
  2. (2y)4⋅(2y){\left(\frac{2}{y}\right)}^{4}\cdot \left(\frac{2}{y}\right)
  3. t3⋅t6⋅t5{t}^{3}\cdot {t}^{6}\cdot {t}^{5}

Answer:

  1. k15{k}^{15}
  2. (2y)5{\left(\frac{2}{y}\right)}^{5}
  3. t14{t}^{14}

The following video gives more examples of using the power rule to simplify expressions with exponents. https://youtu.be/VjcKU5rA7F8

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