document.write( "Question 1032243: 1. Create different sets of expressions that satisfy each desired outcome:\r
\n" ); document.write( "\n" ); document.write( "a. Find a pair of complex binomials whose sum is 8 + 9i.\r
\n" ); document.write( "\n" ); document.write( "b. Find a pair of complex binomials whose difference is -2 + 6i.\r
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\n" ); document.write( "\n" ); document.write( "2. Make a generalization about the exponents of imaginary numbers, regarding whether or not all even or all odd exponents result in real or imaginary numbers. Explain and prove your statement with actual examples and at least one complete sentence that supports your generalization.\r
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\n" ); document.write( "\n" ); document.write( "(Note: Please show all work so that I can understand your logic. Thank you) \n" ); document.write( "

Algebra.Com's Answer #647045 by KMST(5328) $\"\"$ $\"About$

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1.a.
\n" ); document.write( " $\"%282%2B10i%29\"$ and $\"%286-i%29\"$ are two (a pair of) complex binomials that add up to $\"8%2B9i\"$ .
\n" ); document.write( " $\"%282%2B10i%29\"$ $\"%22%2B%22\"$ $\"%286-i%29\"$ $\"%22=%22\"$ $\"8%2B9i\"$ is an option
\n" ); document.write( "because $\"2%2B6=8\"$ and $\"10%2B%28-1%29=9\"$ .
\n" ); document.write( " $\"%283%2B7i%29\"$ $\"%22%2B%22\"$ $\"%285%2B2i%29\"$ $\"%22=%22\"$ $\"8%2B9i\"$ is an option
\n" ); document.write( "because $\"3%2B5=8\"$ and $\"7%2B2=9\"$ .
\n" ); document.write( "The possibilities are endless. All we have to do is find a pair of real numbers that add up to $\"8\"$ for the first term of the two complex binomials, and another pair of real numbers that add up to $\"9\"$ for the coefficients of $\"i\"$ of the two complex binomials.
\n" ); document.write( "THE LOGIC:
\n" ); document.write( " $\"%282%2B10x%29\"$ and $\"%286-x%29\"$ are two binomials that add up to $\"8%2B9x\"$ .
\n" ); document.write( "Those binomials include the (unknown) variable $\"x\"$ ,
\n" ); document.write( "and unless told otherwise we think of $\"x\"$ as a real number.
\n" ); document.write( "Placing the imaginary number $\"i\"$ in the spot where the variable $\"x\"$ was does not change the way we add or subtract. (In math, new definitions are crafted so as to change the way we operate as little as possible. That way, fractional exponents, complex numbers, and the like do not cause us much trouble).
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\n" ); document.write( "1.b. $\"%282%2B10i%29\"$ and $\"%28-4-4i%29\"$ are two (a pair of) complex binomials whose difference is $\"-2%2B6i\"$ .
\n" ); document.write( " $\"%282%2B10i%29\"$ $\"%22-%22\"$ $\"%28-4-4i%29\"$ $\"%22=%22\"$ $\"-2%2B6i\"$ is an option
\n" ); document.write( "As for the sum, the possibilities for the difference are endless. All we have to do is find a pair of real numbers whose difference is $\"-2\"$ for the first term of the two complex binomials, and another pair of real numbers whose difference is $\"62\"$ for the coefficients of $\"i\"$ of the two complex binomials.
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\n" ); document.write( "2. The definition of an imaginary number is usually a complex number like $\"i\"$ , or $\"3i\"$ , or $\"-2i\"$ .
\n" ); document.write( "\"Make a generalization\" and \"Explain and prove your statement with actual examples and at least one complete sentence\" suggest that you are expected to try examples with different odd and even exponents, and maybe even different imaginary numbers until you come up with an \"experimental\" generalization.
\n" ); document.write( "STATEMENT: Powers of imaginary numbers are real whenever the exponent is even, and imaginary whenever the exponent is odd.
\n" ); document.write( "EXAMPLES, \"PROOF\" AND COMPLETE SENTENCES:
\n" ); document.write( " $\"%283i%29%5E2=%283i%29%283i%29=3%2A3%2Ai%2Ai=3%5E2%2Ai%5E2=3%28-1%29=-3\"$ is real, and
\n" ); document.write( " $\"%284i%29%5E2=%284i%29%284i%29=4%2A4%2Ai%2Ai=4%5E2%2Ai%5E2=16%28-1%29=-16\"$ ,
\n" ); document.write( "just like $\"i%5E2=-1\"$ .
\n" ); document.write( " $\"%283i%29%5E3=%283i%29%283i%29%283i%29=3%2A3%2A3%2Ai%2Ai%2Ai=3%5E3%2Ai%5E3=27%28-i%29=-27i\"$ is imaginary,
\n" ); document.write( "just like $\"i%5E3=-i\"$ is imaginary.
\n" ); document.write( "So, the coefficient of $\"i\"$ could be $\"3\"$ or $\"4\"$ or any real number,
\n" ); document.write( "but it makes no difference,
\n" ); document.write( "because the power is real when the same power of $\"i\"$ is real,
\n" ); document.write( "and the power is imaginary when the same power of $\"i\"$ is imaginary.\r
\n" ); document.write( "\n" ); document.write( " $\"i%5E6=i%5E%28%282%2A3%29%29=%28i%5E2%29%5E3=%28-1%29%5E3=-1\"$ and $\"i%5E8=i%5E%28%282%2A4%29%29=%28i%5E2%29%5E4=%28-1%29%5E64=1\"$ Powers of $\"i\"$ with an even exponent can be written as powers of $\"-1\"$ , which is a real number, and that makes them real numbers.
\n" ); document.write( " $\"i%5E7=i%5E%28%286%2B1%29%29=i%5E6%2Ai%5E1=i%5E6%2Ai=%28-1%29%2Ai=-i\"$ and $\"i%5E9=i%5E%28%288%2B1%29%29=i%5E8%2Ai%5E1=i%5E8%2Ai=%281%29%2Ai=i\"$
\n" ); document.write( "Powers of $\"i\"$ with an odd exponent can be written as powers of $\"i\"$ with an even exponent times $\"i\"$ , and since the power with an even exponent is real, its product time $\"i\"$ is imaginary.
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\n" ); document.write( "NOTES:
\n" ); document.write( "Hopefully the teacher would be satisfied with that. Sometimes it is hard to know what kind of answer is expected.
\n" ); document.write( "I do not know if you will understand my logic, but I understand why someone would need help with this problem. This problem may have been created by someone who believes math is an experimental science. I do not agree, but all kinds of beliefs are allowed.
\n" ); document.write( "Or maybe the idea is that some students need to work with concrete examples, before generalizing to abstract concepts.
\n" ); document.write( "Either way, if you are an abstract thinker, you will find this problem annoying, but humoring the teacher is often the best strategy. \n" ); document.write( "

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