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1.4: Exponents

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    87276

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    1.4.1 Learning Objectives

    • Read, interpret, and write exponential notation

    Exponential Notation

    Definition: Exponential Notation

    Just as multiplication is a description of repeated addition, exponen­tial notation is a description of repeated multiplication.

    Suppose we have the repeated multiplication \(8 \cdot 8 \cdot 8 \cdot 8 \cdot 8\). 

    The factor 8 is repeated 5 times.

    Definition: Exponent

    Exponential notation uses a superscript for the number of times the factor is repeated. The superscript is placed on the repeated factor, \(8^5\), in this case. The superscript is called an exponent. An exponent records the number of identical factors that are repeated in a multiplication.

    If \(x\) is any real number and \(n\) is a natural number, then
    \(x^{n}=\underbrace{x \cdot x \cdot x \cdot \ldots \cdot x}_{n \text { factors of } x}\)
    An exponent records the number of identical factors in a multiplication.

    Example 1

    Write the following multiplications using exponents.

    1. \(3 \cdot 3\)
    2. \(62 \cdot 62 \cdot 62 \cdot 62 \cdot 62 \cdot 62 \cdot 62 \cdot 62 \cdot 62\)

    Solution

    1. Since the factor 3 appears 2 times, we record this as \(3^2\).
    2. Since the factor 62 appears 9 times, we record this as \(62^9\).

     

    Example 3

    Expand (write without exponents) each number.

    1. \(12^4\)
    2. \(706^3\)
    3. \(15^1\)

    Solution

    1. The exponent 4 is recording 4 factors of 12 in a multiplication. Thus, \(12^4 = 12 \cdot 12 \cdot 12 \cdot 12\).
    2. The exponent 3 is recording 3 factors of 706 in a multiplication. Thus, \(706^3 = 706 \cdot 706 \cdot 706\).
    3. The exponent 1 is recording 1 factor of 15 in a multiplication. Thus, \(15^1 = 15\).

    Try It Now 1

    Write the following using exponents.

    \(37 \cdot 37\)

    Answer

    \(37^2\)

    Reading Exponential Notation
    Base Exponent Power

    In \(x^n\),

    \(x\) is the base
    \(n\) is the exponent
    The number represented by \(x^n\) is called a power

    The term \(x^n\) is read as "\(x\) to the \(n\)th."

    In a number such as \(8^5\).

    Base
    8 is called the base.

    Exponent, Power
    5 is called the exponent, or power. \(8^5\) is read as "eight to the fifth power," or more simply as "eight to the fifth," or "the fifth power of eight."

    Squared
    When a number is raised to the second power, it is said to be squared. The number \(5^2\) can be read as

    5 to the second power, or
    5 to the second, or
    5 squared.

    Cubed
    When a number is raised to the third power, it is said to be cubed. The number \(5^3\) can be read as

    5 to the third power, or
    5 to the third, or
    5 cubed.

    When a number is raised to the power of 4 or higher, we simply say that that number is raised to that particular power. The number \(5^8\) can be read as

    5 to the eighth power, or just
    5 to the eighth.

    This page titled 1.4: Exponents is shared under a CC BY license and was authored, remixed, and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.