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5.1: Definition of aⁿ

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    45181

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    Definition: \(a^n\)

    For any real number \(a\) and a positive number \(n\), \(a^n\) is the repeated multiplication of \(a\) by itself \(n\) times.

    \[a^n= a\cdot a \cdot a\cdot a\cdot a\cdot a\cdot a\cdot a \ldots \ldots \cdot a \nonumber \]

    Notation:

    \(a\) is the base, \(n\) is the positive exponent.

    \(a^n\) is read as ”\(a\) raised to the power of \(n\).”

    Example 5.1.1

    Identifying the base and exponent in expressions.

    \(2^4\), \(x^5\), \(\left(\dfrac{3}{7}\right)^7\), \((-3)^3\)

    Solution
    Expression Base Exponent
    \(2^4\) 2 4
    \(x^5\) \(x\) 5
    \(\left(\dfrac{3}{7}\right)^7\) \(\dfrac{3}{7}\) 7
    \((-3)^3\) -3 3
    Practice Problem

    Identify the base and exponent of the following.

    Expression Base Exponent
    \(7^9\)
    \((-11)^6\)
    \(a^b\)
    \(\left(\dfrac{11}{12}\right)^5\)
    \(12^3\)
    \(\left(-\dfrac{7}{3}\right)^2\)
    \(x^7\)
    \((2.56)^4\)

    Evaluating expressions of the form \(a^n\)

    When the base and exponent is a numerical value it is possible to evaluate an expression written in with an exponent. To find the value, use the definition and expand the expression. Once expanded, multiply and the result is the numerical value of the expression.

    Example 5.1.2

    Expand the following expressions and evaluate if possible.

    \(3^4\), \(\left(\dfrac{3}{5}\right)^3\), \(x^7\), \((3.12)^2\), \((-5)^3\), \((-y)^6\)

    Solution
    \(3^4\) \(= 3\cdot 3\cdot 3\cdot 3 = 81\)
    \(\left(\dfrac{3}{5}\right)^3\) \(\dfrac{3 }{5} \cdot \dfrac{3}{ 5 }\cdot \dfrac{3 }{5} = \dfrac{27 }{125}\)
    \(x^7\)

    \(x\cdot x\cdot x\cdot x\cdot x\cdot x\cdot x\)

    Note: Can’t evaluate since x is unknown
    \((3.12)^2\) \((3.12)\cdot (3.12) = 9.734\)
    \((-5)^3\) \(−5 \cdot −5 \cdot −5 = −12\)
    \((-y)^6\)

    \(−y \cdot −y \cdot −y \cdot −y \cdot −y \cdot −y = y^6\)

    Note: y is unknown
    Exercise 5.1.1

    Expand the following expressions and evaluate if possible.

    1. \(7^3\)
    2. \(\left(−\dfrac{ 2 }{3}\right)^4\)
    3. \((−x)^7\)
    4. \((7.14)^2\)
    5. \((−3)^9\)
    6. \((z)^5\)
    7. \(\left(− \dfrac{11 }{33 }\right)^2\)
    8. \(6^5\)
    9. \(\left(\dfrac{x}{ y}\right)^4\)
    10. \(a^{10}\)
    11. \(\left(\dfrac{2}{x}\right)^3\)

    This page titled 5.1: Definition of aⁿ is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Victoria Dominguez, Cristian Martinez, & Sanaa Saykali (ASCCC Open Educational Resources Initiative) .