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Math Help - Adding Exponents

  1. #1
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    Adding Exponents

    I'm fairly clear on the rules govening the multiplication of exponents with the same base, but what about adding them?

    I.E. 2^5 + 2^5 + 3^6 + 3^6 + 3^6 = ?

    The first part 2^5 + 2^5 is 2^6 because one is adding one 2^5 to another 2^5

    but using this logic would 3^5 + 3^5 + 3^5 equal? 3^7?

    The latter is incorrect, it should be 3^6? but why?
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  2. #2
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    It's because you need to see repeated addition as multiplication.

    2^5 + 2^5 is the same as "2 lots of 2^5".

    2(2^5) = (2^1)(2^5) = 2^(1 + 5) = 2^6.


    For the second, you have 3^5 + 3^5 + 3^5, in other words "3 lots of 3^5".

    3(3^5) = (3^1)(3^5) = 3^(1 + 5) = 3^6.
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  3. #3
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    The base is 2 in the first, adding 2 numbers and 3 in the second adding 3 numbers.
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    Thank Prove it..

    So does that mean if we have 2^5 + 2^5 + 2^5 + 2^5 , it would be 2^8 as there are four lots of 2?

    2.2.2(2^5) = 2^2+^6 = 2^8?
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    Correct
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    Hello, dumluck!

    2^5 + 2^5 + 3^6 + 3^6 + 3^6 = ?

    The first part 2^5 + 2^5 is 2^6 because one is adding one 2^5 to another 2^5

    but using this logic would 3^5 + 3^5 + 3^5 equal 3^7 ?

    The latter is incorrect, it should be 3^6? but why?

    Here's a baby-talk approach which should clear away any fog . . .


    How do we add x + x ?

    Factor: .x + x .= .x(1 + 1) .= .x(2) .= .2x


    Then: .3^5 + 3^5 + 3^5 .= .3^5(1 + 1 + 1) .= .(3^5)(3) .= .3^6


    And: .2^5 + 2^5 + 2^5 + 2^5 .= .2^5(1 + 1 + 1 + 1)

    . . . . = .(2^5)(4) .= .(2^5)(2^2) .= .2^7

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    Quote Originally Posted by Soroban View Post
    Hello, dumluck!


    Here's a baby-talk approach which should clear away any fog . . .


    How do we add x + x ?

    Factor: .x + x .= .x(1 + 1) .= .x(2) .= .2x


    Then: .3^5 + 3^5 + 3^5 .= .3^5(1 + 1 + 1) .= .(3^5)(3) .= .3^6


    And: .2^5 + 2^5 + 2^5 + 2^5 .= .2^5(1 + 1 + 1 + 1)

    . . . . = .(2^5)(4) .= .(2^5)(2^2) .= .2^7

    So I was wrong above ; 2^5 + 2^5 + 2^5 + 2^5 = (2^5)(4) = (2^5)(2)(2) = (2^5)(2^2) ; when you multiply exponents you add the exponent so it equals 2^7

    what about this example?

    So 3^6 + 3^6 + 3^6 + 3^6 = 4(3^6) ????

    I would assume therefore that you cannot add exponents with different bases, the same way you cannot multiply them?
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    [QUOTE=dumluck;643221]So 3^6 + 3^6 + 3^6 + 3^6 = 4(3^6) ????
    QUOTE]

    ok, i'll at least try this!!... 3^2.3^1.3^6 , is it therefore 3^9?
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    [QUOTE=dumluck;643247]
    Quote Originally Posted by dumluck View Post
    So 3^6 + 3^6 + 3^6 + 3^6 = 4(3^6) ????
    QUOTE]

    ok, i'll at least try this!!... 3^2.3^1.3^6 , is it therefore 3^9?
    No, you were right the first time. (4)(3^6) = (2^2)(3^6). We cannot simplify this, in general.

    Note that if the bases are equal we can apply (a^n)(a^m) = a^(n + m)
    If the exponents are equal we can apply (a^n)(b^n) = (ab)^n

    But we have neither situation in this case.

    -Dan
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  10. #10
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    [QUOTE=topsquark;643252]
    Quote Originally Posted by dumluck View Post
    No, you were right the first time. (4)(3^6) = (2^2)(3^6). We cannot simplify this, in general.

    Note that if the bases are equal we can apply (a^n)(a^m) = a^(n + m)
    If the exponents are equal we can apply (a^n)(b^n) = (ab)^n

    But we have neither situation in this case.

    -Dan
    Hi Dan,

    So for example (9)(3^4) would be ok as it can be broken into 3^2.3^1? so 9 additions of 3^4 would be 3^7?
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  11. #11
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    [QUOTE=dumluck;643256]
    Quote Originally Posted by topsquark View Post

    Hi Dan,

    So for example (9)(3^4) would be ok as it can be broken into 3^2.3^1? so 9 additions of 3^4 would be 3^7?
    Close.


    That's the second time you've included an extra factor. I'm not sure why you are doing it, but at least you are doing it consistently. Notice that 9 is not equal to 3^2 * 3^1 = 27. I'm not sure why you are including the 3^1 in there.

    -Dan
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    [QUOTE=topsquark;643309]
    Quote Originally Posted by dumluck View Post
    Close.


    That's the second time you've included an extra factor. I'm not sure why you are doing it, but at least you are doing it consistently. Notice that 9 is not equal to 3^2 * 3^1 = 27. I'm not sure why you are including the 3^1 in there.

    -Dan
    fatigue . I think I'm ok. (9)(3^6) = (3^6 + 3^6 + 3^6 + 3^6 + 3^6 + 3^6 + 3^6 + 3^6 + 3^6) = (3^2)(3^6) = (3^8) is that right?
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  13. #13
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    [QUOTE=dumluck;643327]
    Quote Originally Posted by topsquark View Post

    fatigue . I think I'm ok. (9)(3^6) = (3^6 + 3^6 + 3^6 + 3^6 + 3^6 + 3^6 + 3^6 + 3^6 + 3^6) = (3^2)(3^6) = (3^8) is that right?
    Yup.

    -Dan
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