{"id":1253,"date":"2023-02-02T12:10:34","date_gmt":"2023-02-02T12:10:34","guid":{"rendered":"https:\/\/www.storyofmathematics.com\/?page_id=1253"},"modified":"2023-02-24T12:58:06","modified_gmt":"2023-02-24T12:58:06","slug":"negative-exponents","status":"publish","type":"page","link":"https:\/\/www.storyofmathematics.com\/negative-exponents\/","title":{"rendered":"Negative Exponents \u2013 Explanation & Examples"},"content":{"rendered":"\t\t
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Negative Exponents \u2013 Explanation & Examples<\/h1>

\"NegativeExponents are powers or indices. An exponential expression consists of two parts, namely the base, denoted as b and the exponent, denoted as n. The general form of an exponential expression is b n<\/sup>. For example, 3 x 3 x 3 x 3 can be written in exponential form as 34<\/sup> where 3 is the base and 4 is the exponent. They are widely used in algebraic problems, and for this reason, it is important to learn them so as to make the studying algebra easy.<\/p>

Many students will find it hard to understand negative numbers and fractions. It normally a total disaster when negative exponents are added to the equations. Well, not really. Learning negative exponents is a major foundation block for solving advanced mathematical expressions. This is because, it equips students with the necessary skills and knowledge to face challenging problems in and out of the classroom.<\/p>

If you are wondering where to begin, don\u2019t worry, this article is going to help you transform your course on negative exponents into a positive experience.\n

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To help you understand the negative exponent rule better, this paper discusses in detail the following topics of negative exponent rule:<\/em><\/p>

  • Negative exponents rule<\/li>
  • Examples of negative exponents<\/li>
  • Negative fractional exponents<\/li>
  • How to solve Fractions with negative exponents<\/li>
  • How to multiply negative exponents<\/li>
  • Dividing negative exponents<\/li><\/ul>

    Before we tackle each one of these topics, let us do a quick recap of the rules of exponents.<\/em><\/p>

    • Multiplication of powers with same base: With multiplication of like bases, add the powers together.<\/li>
    • Quotient of powers rule: When dividing like bases, the powers are subtracted<\/li>
    • Power of powers rule: Multiply powers together when raising a power by another exponent<\/li>
    • Power of a product rule: Distribute power to each base when raising several variables by a power<\/li>
    • Power of a quotient rule: Distribute power to each base when raising several variables by a power<\/li>
    • Zero power rule: This rule implies that, any base raised to a power of zero is equal to one<\/li>
    • Negative exponent rule: To convert a negative exponent to a positive one, write the number into a reciprocal.<\/li><\/ul>

      How to Solve Negative Exponents?<\/h2>

      The law of negative exponents states that, when a number is raised to a negative exponent, we divide 1 by the base raised to a positive exponent. The general formula of this rule is: a -m<\/sup> = 1\/a m <\/sup>and (a\/b) -n<\/sup> = (b\/a) n<\/sup>.\"How<\/p>

      Example 1<\/em><\/strong><\/p>

      Below are examples of how negative exponent rule works:<\/p>

      • 2 -3<\/sup>= 1\/2 3<\/sup>\u00a0= 1\/ (2 x 2 x 2) = 1\/8 = 0.125<\/li>
      • 2 -2<\/sup> = 1\/2 2<\/sup> = 1\/4<\/li>
      • (2\/3) -2 <\/sup>= (3\/2) 2<\/sup><\/li><\/ul>

        Negative fractional exponents<\/h2>

        The base b raised to the negative power of n\/m is equivalent to 1 divided by the base b raised to the positive exponent of n\/m:<\/p>

        b -n\/m<\/sup>\u00a0= 1 \/\u00a0b n\/m<\/sup>\u00a0= 1 \/\u00a0(m <\/sup>\u221ab) n<\/sup><\/p>

        It implies that, if the base 2 is raised to the negative exponent of 1\/2, it is equivalent to 1 divided by the base 2 raised to the positive exponent of 1\/2:<\/p>

        2-1\/2<\/sup>\u00a0= 1\/21\/2<\/sup>\u00a0= 1\/\u221a<\/em>2\u00a0= 0.7071<\/p>

        You should notice that a fractional negative exponent is the same as finding the root of the base.<\/p>

        Fractions with negative exponents<\/h2>

        The rule implies that, if a fraction a\/b is raised to the negative exponent of n, it is equal to 1 divided by the base a\/b raised to the positive exponent of n:<\/p>

        (a\/b) -n<\/sup>\u00a0= 1 \/ (a\/b) n<\/sup>\u00a0= 1 \/ (a n<\/sup>\/b n<\/sup>) =\u00a0b n<\/sup>\/a n<\/sup><\/p>

        The base 2\/3 raised to the negative exponent of 2 is equal to 1 divided by the base 2\/3 raised to the positive exponent of 2. In other words, 1 is divide by the reciprocal of the base raised to a positive exponent of 2<\/p>

        (2\/3) -2<\/sup>\u00a0= 1 \/ (2\/3) 2<\/sup>\u00a0= 1 \/ (2 2<\/sup>\/3 2<\/sup>) = (3\/2)2\u00a0<\/sup>= 9\/4 = 2.25<\/p>

        Multiplication of negative exponents<\/h3>

        When exponents with the same base are multiplied, we can add the exponents:<\/p>

        a\u00a0-n<\/sup>\u00a0x a\u00a0-m<\/sup>\u00a0=\u00a0a\u00a0-(n + m<\/sup>)\u00a0=<\/sup> 1 \/\u00a0a\u00a0n + m<\/sup><\/p>

        Example 2<\/em><\/strong><\/p>

        2 -3<\/sup>\u00a0x 2 -4<\/sup>\u00a0= 2 -(3 + 4)<\/sup>\u00a0= 2 -7<\/sup>\u00a0= 1 \/ 2 7<\/sup>\u00a0= 1 \/ (2 x 2 x 2 x 2 x 2 x 2 x 2) = 1 \/ 128 = 0.0078125<\/p>

        In the case of different bases and common exponents of a and b, we can multiply a and b:<\/p>

        a\u00a0-n<\/sup>\u00a0\u22c5\u00a0b\u00a0-n<\/sup>\u00a0= (a\u00a0\u22c5\u00a0b)\u00a0-n<\/sup><\/p>

        Example 3<\/em><\/strong><\/p>

        3 -2<\/sup>\u00a0x 4 – 2<\/sup>\u00a0= (3 x 4) -2<\/sup>\u00a0= 12 -2<\/sup>\u00a0= 1 \/ 12 2<\/sup>\u00a0= 1 \/ (12 x 12) = 1 \/ 144 = 0.0069444<\/p>

        In case both the bases and the exponents are different we calculate each exponent separately and then multiply:<\/p>

        a\u00a0-n<\/sup><\/em>\u00a0\u22c5\u00a0b\u00a0-m<\/sup><\/em><\/p>

        Example 4<\/em><\/strong><\/p>

        3-2<\/sup>\u00a0x 4-3<\/sup>\u00a0= (1\/9) x (1\/64) = 1 \/ 576 = 0.0017361<\/p>

        How to divide negative exponents<\/p>

        In the case of exponents with the same base, we subtract the exponents:<\/p>

        a\u00a0-n<\/sup>\u00a0\/\u00a0a–\u00a0m<\/sup>\u00a0=\u00a0a\u00a0-n + m<\/sup><\/p>

        Example 5<\/em><\/strong><\/p>

        2 -6<\/sup>\/2 -3<\/sup> = 2 -6+3<\/sup><\/p>

        = 2-3<\/sup><\/p>

        = 1\/23<\/sup><\/p>

        = 1\/8<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t

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        Practice Questions<\/h3>