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Math Help - Radical simplification

  1. #1
    Member MathBlaster47's Avatar
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    Radical simplification

    Here are some questions that I have been struggling with for quite some time, mostly because I don't understand how to begin to work them out.

    1:
    Simplify if possible. If the expression cannot be simplified, write "not possible" as the answer.

    \frac{\sqrt{5}+1}{\sqrt{5}-1}

    I'm leaning toward not possible with this one, but it doesn't hurt to make sure.

    2:
    Solve for the unknown and check your work.

    (a):
    \sqrt{y}+\sqrt{y+7}=7
    I used Microsoft math to get the answer, y\approx9.
    However, I've yet to figure out how to reach that answer. I'm guessing that saying: \sqrt{y}=7-\sqrt{y+7} won't really cut it.

    (b):
    \sqrt{x+11}=\sqrt{x}+1
    Same as above, I used Microsoft math to get the final answer, but have not quite worked out how to reach said answer: x=25.
    From looking at it I can see why 25 is the answer, but I don't think anyone would accept "it looks like 25 is a good answer"--even if I'm correct.

    Thank you!
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  2. #2
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    Quote Originally Posted by MathBlaster47 View Post
    Here are some questions that I have been struggling with for quite some time, mostly because I don't understand how to begin to work them out.

    1:
    Simplify if possible. If the expression cannot be simplified, write "not possible" as the answer.

    \frac{\sqrt{5}+1}{\sqrt{5}-1}

    I'm leaning toward not possible with this one, but it doesn't hurt to make sure.

    2:
    Solve for the unknown and check your work.

    (a):
    \sqrt{y}+\sqrt{y+7}=7
    I used Microsoft math to get the answer, y\approx9.
    However, I've yet to figure out how to reach that answer. I'm guessing that saying: \sqrt{y}=7-\sqrt{y+7} won't really cut it.

    (b):
    \sqrt{x+11}=\sqrt{x}+1
    Same as above, I used Microsoft math to get the final answer, but have not quite worked out how to reach said answer: x=25.
    From looking at it I can see why 25 is the answer, but I don't think anyone would accept "it looks like 25 is a good answer"--even if I'm correct.

    Thank you!
    rationalize the denominator ...

    \frac{\sqrt{5}+1}{\sqrt{5}-1} \cdot \frac{\sqrt{5}+1}{\sqrt{5}+1}

    multiply and simplify.




    \sqrt{y}+\sqrt{y+7}=7

    \sqrt{y+7}=7 - \sqrt{y}

    square both sides ...

    y+7 = 49 - 14\sqrt{y} + y

    14\sqrt{y} = 42

    \sqrt{y} = 3<br />

    y = 9

    do the second radical equation in the same manner.
    Last edited by skeeter; February 27th 2010 at 08:55 AM. Reason: fix signs
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  3. #3
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    Quote Originally Posted by MathBlaster47 View Post
    Here are some questions that I have been struggling with for quite some time, mostly because I don't understand how to begin to work them out.

    1:
    Simplify if possible. If the expression cannot be simplified, write "not possible" as the answer.

    \frac{\sqrt{5}+1}{\sqrt{5}-1}

    I'm leaning toward not possible with this one, but it doesn't hurt to make sure.

    2:
    Solve for the unknown and check your work.

    (a):
    \sqrt{y}+\sqrt{y+7}=7
    I used Microsoft math to get the answer, y\approx9.
    However, I've yet to figure out how to reach that answer. I'm guessing that saying: \sqrt{y}=7-\sqrt{y+7} won't really cut it.

    (b):
    \sqrt{x+11}=\sqrt{x}+1
    Same as above, I used Microsoft math to get the final answer, but have not quite worked out how to reach said answer: x=25.
    From looking at it I can see why 25 is the answer, but I don't think anyone would accept "it looks like 25 is a good answer"--even if I'm correct.

    Thank you!
    1. Depends how you define "simplify". There are no common factors but it's convention to rationalise the denominator by multiplying it by it's conjugate.

    In this case multiply by \frac{\sqrt5 +1}{\sqrt5 +1}


    2. You'll have to square it twice and check for extraneous solutions

    (\sqrt{y} + \sqrt{y+7})^2 = 7^2

    y + 2\sqrt{y(y+7)}+y+7 = 49

    2\sqrt{y^2+7y} = 42-2y

    Square again

    4(y^2+7y) = 42^2- 2 \cdot 42 \cdot 2y + 4y^2

    Simplify and then you have a quadratic in y. Be sure to check for extraneous solution
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  4. #4
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    Quote Originally Posted by MathBlaster47 View Post
    Simplify if possible. If the expression cannot be simplified, write "not possible" as the answer.
    \frac{\sqrt{5}+1}{\sqrt{5}-1}
    I'm leaning toward not possible with this one, but it doesn't hurt to make sure.
    CAUTION: one does not "lean toward" in maths

    Multiply numerator and denominator by SQRT(5)+1

    You'll get [SQRT(5)+1]^2 / 4

    Expand numerator:
    [6 + 2SQRT(5)] / 4

    Simplify numerator:
    2[3 + SQRT(5)] / 4

    Wrap up:
    [3 + SQRT(5)] / 2
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  5. #5
    Member MathBlaster47's Avatar
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    I get it now, thanks guys!
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  6. #6
    Member MathBlaster47's Avatar
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    Uhm....I just noticed something as I tried to work out, sqrt{x+11}=sqrt{x}+1...I am a little confused. If I square both sides of the equation, I make it false, I have nothing to add or subtract from both sides of the equation either.....what do I do now?
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  7. #7
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    Quote Originally Posted by MathBlaster47 View Post
    Uhm....I just noticed something as I tried to work out, \sqrt{x+11}=\sqrt{x}+1...I am a little confused. If I square both sides of the equation, I make it false, I have nothing to add or subtract from both sides of the equation either.....what do I do now?
    squaring both sides ...

    x+11 = x + 2\sqrt{x} + 1

    keep going.
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