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5.2: The Product Rule for Exponents

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    Definition: The Product Rule for Exponents

    For any real number \(a\) and positive numbers \(m\) and \(n\), the product rule for exponents is the following.

    \(a^m \cdot a^n = a^{m+n}\)

    Note: Bases must be the same to use the product rule.

    Idea:

    From the last section, \(x^3 = \textcolor{blue}{ x \cdot x \cdot x }\qquad x^5 = \textcolor{red}{x \cdot x \cdot x \cdot x \cdot x}\)

    Their product

    \(x^3 \cdot x^5 = \textcolor{blue}{x \cdot x \cdot x} \textcolor{red}{\cdot x \cdot x \cdot x \cdot x \cdot x} = x^8\)

    Hence, \(x^3 \cdot x^5 = x^{3+5 }= x^8\)

    Example 5.2.1

    Use the product rule of exponents to simplify expressions.

    1. \(k^3 \cdot k^9\)
    2. \(\left(\dfrac{2 }{7}\right)^2 \cdot \left(\dfrac{2 }{7}\right)^6\)
    3. \((−2a)^3 \cdot (−2a)^7\)
    4. \(x \cdot x^3 \cdot x^{11}\)
    5. \(y^{13 }\cdot y^{33}\)
    6. \(x^3 \cdot y^2 \cdot x \cdot y^4\)
    Solution
    Expression Product Rule Base
    \(k^3 \cdot k^9\) \(k^{3+9}= k^{12}\) \(k\)
    \(\left(\dfrac{2 }{7}\right)^2 \cdot \left(\dfrac{2 }{7}\right)^6\) \(\left( \dfrac{2 }{7}\right)^{2+6 }= \left(\dfrac{2 }{7}\right)^8\) \(\dfrac{2}{7}\)
    \((−2a)^3 \cdot (−2a)^7\) \((−2a)^{3+7 }= (−2a)^{10}\) \(-2a\)
    \(x \cdot x^3 \cdot x^{11}\) \(x ^{1+3+11 }= x^{15}\) \(x\)
    \(y^{13 }\cdot y^{33}\) \(y^{13+33 }= y^46\) \(y\)
    \(x^3 \cdot y^2 \cdot x \cdot y^4\) \(x^{3+1 }\cdot y ^{2+4 }= x^{ 4 }\cdot y^{6}\) \(x\) and \(y\)

    Note: Again, the bases MUST be the same to simplify using the product rule of exponent

    Helpful steps to simplify using the product rule of exponents:

    1. Identify terms with common bases
    2. Identify the exponent of common bases.
    3. Add exponents of common bases and make the result of the sum the new exponent.
    4. Repeat steps as need
    Exercise 5.2.1

    Use the product rule of exponents to simplify the following.

    1. \(f^3 \cdot f^11\)
    2. \(\left(\dfrac{x}{7}\right)^2 \cdot \left(\dfrac{x }{7}\right)^3\)
    3. \((−7x)^9 \cdot (−7x)^7\)
    4. \(h^5 \cdot h^3 \cdot h^{11}\)
    5. \(t^{13} \cdot t^{33}\)
    6. \(x^8 \cdot y^2 \cdot z \cdot x^ 3 \cdot y^2 \cdot z^{17}\)
    7. \(x^3 \cdot y^4 \cdot x^3\)

    This page titled 5.2: The Product Rule for Exponents was last modified on Mon, 18 Jul 2022 02:14:35 GMT and is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Victoria Dominguez, Cristian Martinez, and Sanaa Saykali (ASCCC Open Educational Resources Initiative) .