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06-19 MESA Challenge Math Analysis

(4 Questions)
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MESA Challenge set for Math Analysis
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i^{4} + i^{7} + i^{2} = ?
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Find the result of this expression: (i^{2}+3i+2)(\overline{7+4i}) . Remember that the conjugate of an expression is represented by a line over the original expression. Enter the resulting complex number with the real number in front.
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A complex number is a number that contains both real and imaginary components. Real numbers are the normal numbers you’ve been working with already, but imaginary numbers are essentially multiples of “i” where i = \sqrt{-1}. i is often referred to as the imaginary unit. What is i^{2}? Hint: Think about what happens when you square a square root.
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You will likely encounter expressions where the imaginary number is in the denominator. It is likely that you will need to make the denominator solely consist of real numbers. In order to do this, we multiply both the numerator and the denominator by the complex conjugate, which is essentially the imaginary denominator but with the opposite arithmetic sign. For example, the complex conjugate of 2+3i is simply 2-3i. Note that conjugates are denoted by a line over the expression, so 2-3i =\overline{2+3i} Knowing this, rewrite the following expressions with a real denominator, then give the sum of all coefficients (ie (2i+3)/(6) would add up to 11). Separate answers with a comma and no spaces. a. \frac{1}{3-11i} b. \frac{2+i}{2-i}
by Frank T