SOLUTION: 1. Create different sets of expressions that satisfy each desired outcome: a. Find a pair of complex binomials whose sum is 8 + 9i. b. Find a pair of complex binomials whose

Question 1032243: 1. Create different sets of expressions that satisfy each desired outcome:
a. Find a pair of complex binomials whose sum is 8 + 9i.
b. Find a pair of complex binomials whose difference is -2 + 6i.

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.

(Note: Please show all work so that I can understand your logic. Thank you)
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
1.a.
and are two (a pair of) complex binomials that add up to .

is an option
because and .

is an option
because and .
The possibilities are endless. All we have to do is find a pair of real numbers that add up to for the first term of the two complex binomials, and another pair of real numbers that add up to for the coefficients of of the two complex binomials.
THE LOGIC:
and are two binomials that add up to .
Those binomials include the (unknown) variable ,
and unless told otherwise we think of as a real number.
Placing the imaginary number in the spot where the variable 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).

1.b. and are two (a pair of) complex binomials whose difference is .

is an option
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 for the first term of the two complex binomials, and another pair of real numbers whose difference is for the coefficients of of the two complex binomials.

2. The definition of an imaginary number is usually a complex number like , or , or .
"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.
STATEMENT: Powers of imaginary numbers are real whenever the exponent is even, and imaginary whenever the exponent is odd.
EXAMPLES, "PROOF" AND COMPLETE SENTENCES:
is real, and
,
just like .
is imaginary,
just like is imaginary.
So, the coefficient of could be or or any real number,
but it makes no difference,
because the power is real when the same power of is real,
and the power is imaginary when the same power of is imaginary.
and Powers of with an even exponent can be written as powers of , which is a real number, and that makes them real numbers.
and
Powers of with an odd exponent can be written as powers of with an even exponent times , and since the power with an even exponent is real, its product time is imaginary.

NOTES:
Hopefully the teacher would be satisfied with that. Sometimes it is hard to know what kind of answer is expected.
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.
Or maybe the idea is that some students need to work with concrete examples, before generalizing to abstract concepts.
Either way, if you are an abstract thinker, you will find this problem annoying, but humoring the teacher is often the best strategy.
RELATED QUESTIONS
a rectangular box has a surface area of 960 squared centimetres. it has a square end.... (answered by josgarithmetic)

1. Is it possible to generalize a condition for creating any two binomials whose sum is a (answered by stanbon)

A die is weighted so that the even number outcomes are twice as likely as an odd outcome. (answered by greenestamps)

Give an example of the sum of two binomials that is a... (answered by edjones)

1. A store has 6 TV Graphic magazines and 8 Newstime magazines on the counter. If two... (answered by math_tutor2020)

give an order pair(a,b) of numbeers that satisfy the following equation.... (answered by stanbon)

Two consecutive odd integers have a product of 99. Find all sets of integers that satisfy (answered by Alan3354)

Two consecutive odd integers have a product of 99. Find all sets of integers that satisfy (answered by stanbon)

two consecutive odd integers have a product of 99. find all sets of integers that satisfy (answered by stanbon,Alan3354)