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Math Help - What is the clue to think that way?

  1. #1
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    What is the clue to think that way?

    In the "Foundation Maths" book in chapter (19): "The exponential function", there was an example solving the following problem (I used "^" to denote power):

    (e^2t + 2e^2 + 1)^1/2

    In solving the problem, this was converted to ((e^t + 1)^2)^1/2

    What clue can tell that I have to solve the problem this way and not by just taking the root of every element?

    Thanks.
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  2. #2
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    Quote Originally Posted by SWEngineer View Post
    In the "Foundation Maths" book in chapter (19): "The exponential function", there was an example solving the following problem (I used "^" to denote power):

    (e^2t + 2e^2 + 1)^1/2

    In solving the problem, this was converted to ((e^t + 1)^2)^1/2

    What clue can tell that I have to solve the problem this way and not by just taking the root of every element?

    Thanks.
    \sqrt{e^{2t}+2e^2+1}=\sqrt{\left(e^t+1\right)\left  (e^t+1\right)}

    Taking the square root of each term does not work.

    \sqrt{4}+\sqrt{4}=2+2=4

    \sqrt{4+4}=\sqrt{8}=\sqrt{(4)2}=\sqrt{4}\sqrt{2}=2  \sqrt{2}<4

    Or

    \sqrt{8}<\sqrt{9}\Rightarrow\ \sqrt{8}<3<4

    To take a square root, you can do so when you are taking the square root of a square.

    2^2=4\Rightarrow\sqrt{4}=2

    3^2=9\Rightarrow\sqrt{9}=3

    3^2+4^2=5^2\Rightarrow\sqrt{3^2+4^2}=\sqrt{5^2}=5


    \left(e^t+1\right)\left(e^t+1\right)=e^t\left(e^t+  1\right)+1\left(e^t+1\right)=e^{2t}+e^t+e^t+1=e^{2  t}+2e^t+1

    However, this is a square

    e^{2t}+2e^t+1=\left(e^t+1\right)^2
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  3. #3
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    Quote Originally Posted by SWEngineer View Post
    In the "Foundation Maths" book in chapter (19): "The exponential function", there was an example solving the following problem (I used "^" to denote power):

    (e^2t + 2e^2 + 1)^1/2

    In solving the problem, this was converted to ((e^t + 1)^2)^1/2

    What clue can tell that I have to solve the problem this way and not by just taking the root of every element?

    Thanks.
    The "clue" is recognizing perfect squares. x^2+ 2x+ 1= (x+ 1)^2

    Here, x= e^t so x^2= (e^t)^2= e^{2t}
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  4. #4
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    Thanks a lot for your replies. It is clear now.
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