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Understanding exponents

  1. Jun 24, 2014 #1
    Hi,

    I am helping my kid with exponents. I told her that the exponent tells us how many times we should multiply the base number. While it works with a simple example like 4^6, I am not sure how to explain her why 4^0 =1 and why 27^(1/3) = 3.

    Any ideas?

    Thanks.
     
  2. jcsd
  3. Jun 24, 2014 #2

    ehild

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    Show how to multiply and divide powers of the same base -with simple numbers first.

    For example, ##2^3\cdot 2^4= (2\cdot2\cdot2)\cdot(2\cdot2\cdot2\cdot2) = 2^7##. The exponents add. an*am=an+m.

    You can simplify the fraction when dividing powers.

    [tex]\frac{2^4}{2^3}=\frac{2\cdot2\cdot2\cdot2}{2\cdot2\cdot2}=2= 2^{4-3}[/tex].
    [tex]\frac{a^n}{a^m}=a^{n-m}[/tex]



    What happens if n=m?

    [tex]\frac{2^3}{2^3}=\frac{2\cdot2\cdot2}{2\cdot2\cdot2}=1= 2^{3-3}=2^0[/tex]

    a0=1...

    You can show what a negative exponent means:

    [tex]\frac{2^3}{2^4}=\frac{2\cdot2\cdot2}{2\cdot2\cdot2\cdot2}=\frac{1}{2}= 2^{3-4}=2^{-1}[/tex].


    The next is to show how to get the power of a power:

    ##\left(2^3\right)^4=(2\cdot2\cdot2)\cdot(2\cdot2\cdot2)\cdot(2\cdot2 \cdot2)\cdot(2\cdot2\cdot2)= 2^{12}## You multiply the powers. (an)m=anm.

    What does it mean when the power is a fraction, 1/3, for example?

    ## \left(2^{1/3}\right) ^3=2^{\frac{1}{3}\cdot 3}=2^1=2##.

    a1/3 is a number the third power of which is a.

    As for 271/3: 27=33.
    271/3=(33)1/3=33*1/3=31=3

    ehild
     
    Last edited: Jun 24, 2014
  4. Jun 24, 2014 #3

    micromass

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    It all relies on knowing the laws ##a^{m+n} = a^m a^n## This is a fundamental property for understanding exponents.

    Then it is certainly true that ##1+0 = 1##. So if we put those in the exponents, then it must be true that ##4^{1 + 0} = 4^1##. Thus ##4^1 4^0 = 4^1##. Of course, ##4^1 = 4##. Thus we have something like ##4\cdot 4^0 = 4##. So ##4^0## is some number when multiplied by ##4##, it will give ##4## again. We see immediately that ##4^0 = 1##.

    For ##27^{1/3}## something similar holds. Of course we know that ##\frac{1}{3}+ \frac{1}{3} + \frac{1}{3} = 1##. So if we put this in the exponents, we get
    [tex]27^{\frac{1}{3}+ \frac{1}{3} + \frac{1}{3}} = 27^1 = 27[/tex]
    And when using our fundamental property, we see that
    [tex]27^{1/3}27^{1/3}27^{1/3} = 27[/tex]
    or just
    [tex](27^{1/3})^3 = 27[/tex]
    So ##27^{1/3}## is the number such that if we cube it, we get ##27##. But by inspection we see that ##3## is such a number since ##3^3 = 27##, so we must have ##27^{1/3} = 3##.
     
  5. Jun 24, 2014 #4
    Thanks folks.
     
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