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Math Help - complex numbers

  1. #1
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    complex numbers


    Evaluate the expression and write the result in the form .

    The real number a equals ?
    The real number b equals ?
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    Need help with these to:
    The expression

    equals where
    the coefficient C is _______ , the exponent e is______


    Write the following numbers in a+b form:





    Solve the following inequality. Write the answer in interval notation.
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    Quote Originally Posted by badandy328 View Post

    Evaluate the expression and write the result in the form .

    The real number a equals ?
    The real number b equals ?
    Recall that i = \sqrt{-1}, so basically this is a problem in rationalizing the denominator. As in those types of problems, we want to multiply the top and bottom by the "conjugate" of the denominator, in this case called the "complex conjugate."

    \frac{2 + 2i}{1 - 3i} = \frac{2 + 2i}{1 - 3i} \cdot \frac{1 + 3i}{1 + 3i}

    = \frac{(2 + 2i)(1 + 3i)}{(1 - 3i)(1 + 3i)} = \frac{2 + 6i + 2i + 6i^2}{1 + 3i - 3i - 9i^2}

    = \frac{2 + 8i - 6}{1 + 9} = \frac{-4 + 8i}{10}

    =  -\frac{4}{10} + i \frac{8}{10} = -\frac{2}{5} + i \frac{4}{5}

    So a = -\frac{2}{5} and b = \frac{4}{5}.

    -Dan
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    Quote Originally Posted by badandy328 View Post

    equals where
    the coefficient C is _______ , the exponent e is______
    \frac{x^7(2x)^7}{x^2} = \frac{x^22^7x^7}{x^2}

    = 2^7 \frac{x^{7+7}}{x^2} = 128\frac{x^{14}}{x^2}

    = 128x^{14-2} = 128x^{12}

    So C = 128 and e = 12.

    -Dan
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    [quote=badandy328;28053]
    Write the following numbers in a+b form:

    [quote]
    I'm going to FOIL out each pair of factors, but I'm going to factor a "-" from the second term. This is simply for book-keeping: I hate keeping track of negative signs!

    (3 + 5i)(-4 - i)(-1 + 3i) = -(3 + 5i)(4 + i)(-1 + 3i)

    = -(3 + 5i)(-4 + 12i - i + 3i^2) = -(3 + 5i)(-4 + 11i - 3) = -(3 + 5i)(-7 + 11i)

    = -(-21 + 33i - 35i + 55i^2) = -(-21 - 2i - 55) = -(-76 - 2i)

    = 76 + 2i


    Quote Originally Posted by badandy328 View Post
    ((-5 - 3i)^2 + 3)i = (25 + 30i + 9i^2 + 3)i = (28 + 30i - 9)i

    = (19 + 30i)i = 19i + 30i^2 = -30 + 19i

    -Dan
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  6. #6
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    Thank you sooo much. Is there like a site where you plug this stuff in or are you just actually solving them?
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    Quote Originally Posted by badandy328 View Post
    Solve the following inequality. Write the answer in interval notation.
    First look for the critical points. This is where the expression is 0 or where a 0 appears in the denominator. Since we have no denominator, all we need to know is where the expression is 0. So:
    x^2 - x - 30 = 0

    (x - 6)(x + 5) = 0

    So the critical points are at x = -5 and x = 6.

    Now split the real line into intervals and test each interval.
    (-\infty, -5): x^2 - x - 30 > 0 Yes!
    (-5, 6): x^2 - x - 30 < 0 No.
    (6, \infty): x^2 - x - 30 > 0 Yes!

    So the solution set is (-\infty, -5) \cup (6, \infty).

    -Dan
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  8. #8
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    Quote Originally Posted by badandy328 View Post
    Thank you sooo much. Is there like a site where you plug this stuff in or are you just actually solving them?
    No. I'm solving them. Though I admit to checking the answer on my calculator to help speed the process of finding my mistakes before I post.

    -Dan
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  9. #9
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    2 last things. Is this right?
    If you rationalize the denominator of

    then you will get

    where r, s, and n are all positive integers (with no common factors).
    r= -7
    s= 4
    n=-118


    And the last question I have no clue how to do.
    Find all solutions of the equation and express them in the form :
    First input the solution with b<0 here:
    the real number a equals _____ and the real number b equals ______
    Then input the solution with b>0 here:
    the real number a equals _______ and the real number b equals _________
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  10. #10
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    [quote=badandy328;28061]2 last things. Is this right?
    If you rationalize the denominator of

    then you will get

    where r, s, and n are all positive integers (with no common factors).
    r= -7
    s= 4
    n=-118
    [\quote]

    Almost. The conjugate of a\sqrt{b} + c\sqrt{d} is a\sqrt{b} - c\sqrt{d}. You only negate one of the terms. Think of it this way. We are applying the identity: (a + b)(a - b) = a^2 - b^2. This ensures that the square root signs disappear.

    Quote Originally Posted by badandy328 View Post
    And the last question I have no clue how to do.
    Find all solutions of the equation and express them in the form :
    First input the solution with b<0 here:
    the real number a equals _____ and the real number b equals ______
    Then input the solution with b>0 here:
    the real number a equals _______ and the real number b equals _________
    t + 4 + \frac{6}{t} = 0

    Multiply both sides by t:
    t^2 + 4t + 6 = 0 <-- We have to remember to check the solutions and make sure t \neq 0 because the original expression forbids it. As it happens we won't have to worry here.

    Using the quadratic formula:
    t = \frac{-4 \pm \sqrt{4^2 - 4 \cdot 1 \cdot 6}}{2 \cdot 1}

    t = \frac{-4 \pm \sqrt{16 - 24}}{2}

    t = \frac{-4 \pm \sqrt{-8}}{2}

    t = \frac{-4 \pm \sqrt{-4 \cdot 2}}{2}

    t = \frac{-4 \pm 2\sqrt{-2}}{2}

    t = -2 \pm \sqrt{-2}

    t = -2 \pm i \sqrt{2}

    -Dan
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  11. #11
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    Ok, I see how to came to figure that out. Now I'm having trouble figuring out how to answer that question.
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  12. #12
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    Quote Originally Posted by badandy328 View Post
    Ok, I see how to came to figure that out. Now I'm having trouble figuring out how to answer that question.
    Which question?

    -Dan
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    This one:

    And the last question I have no clue how to do.
    Find all solutions of the equation and express them in the form :
    First input the solution with b<0 here:
    the real number a equals _____ and the real number b equals ______
    Then input the solution with b>0 here:
    the real number a equals _______ and the real number b equals _________[/QUOTE]
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  14. #14
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    Quote Originally Posted by badandy328 View Post
    This one:

    And the last question I have no clue how to do.
    Find all solutions of the equation and express them in the form :
    First input the solution with b<0 here:
    the real number a equals _____ and the real number b equals ______
    Then input the solution with b>0 here:
    the real number a equals _______ and the real number b equals _________
    Ummmm...I did post the solution...

    The real number a is -2 for both questions and b is \pm \sqrt{2}.

    -Dan
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