1. What is the sum of the complex numbers: (−9−i)+(−5−i)?
A. −14−2i
B. −14+2i
C. 14−2i
D. 14+2i
2. What is the product of the complex numbers 8i and 5i?
A. 40
B. -40i
C. -40
D. 40i
3. The product(5+i)(5−i) is a real number, 26. What are the factors (5+i) and (5−i) called?
A. imaginary units
B. complex conjugates
C. imaginary numbers
D. complex numbers
4. What is the product of complex conjugates?
A. The product of complex conjugates is the same as the product of opposites.
B. The product of complex conjugates is a sum of two squares and is always a real number.
C. The product of complex conjugates is a difference of two squares and is always a real number.
D. The product of complex conjugates may be written in standard form as a+bi where neither a nor b is zero.
5. Which of the following expressions is equivalent to a^6+b^6
A. (a^3+ib^2)^2
B. (a^3+ib^2) (a^3-ib^2)
C. (a^3-ib^2)^2
D. (a^+b^2) (a^3-b^2)
1. A. -14-2i
2. C. -40
3. B. complex conjugates
4. C. The product of complex conjugates is a difference of two squares and is always a real number.
5. D. (a^3+b^2) (a^3-b^2)
1. The sum of the complex numbers (-9-i) + (-5-i) can be found simply by adding the real parts and the imaginary parts separately.
Real part: (-9) + (-5) = -14
Imaginary part: (-1) + (-1) = -2
Therefore, the sum of the complex numbers is -14 - 2i.
The correct answer is A. -14 - 2i.
2. To find the product of the complex numbers 8i and 5i, you simply multiply them together.
8i * 5i = (8 * 5) * (i * i) = 40 * (-1) = -40
Therefore, the product of the complex numbers is -40.
The correct answer is C. -40.
3. The product (5+i)(5-i) can be expanded using the FOIL method.
(5 + i)(5 - i) = 5 * 5 + 5 * (-i) + i * 5 + i * (-i)
= 25 - 5i + 5i - i^2
= 25 + 0i - i^2
= 25 + 0 + 1
= 26
The factors (5+i) and (5-i) are called complex conjugates.
The correct answer is B. complex conjugates.
4. The product of complex conjugates is a difference of two squares and is always a real number.
The correct answer is C. The product of complex conjugates is a difference of two squares and is always a real number.
5. The expression a^6 + b^6 can be factored using the difference of squares formula.
a^6 + b^6 = (a^3)^2 + (b^3)^2 = (a^3 + b^3)(a^3 - b^3)
Therefore, the expression is equivalent to (a^3 + b^3)(a^3 - b^3).
The correct answer is D. (a^3 + b^3)(a^3 - b^3).
1. To find the sum of two complex numbers, you simply add the real parts and the imaginary parts separately. In this case, we have (-9 - i) + (-5 - i). Adding the real parts, we get -9 + (-5) = -14. Adding the imaginary parts, we get -1 + (-1) = -2. So the sum is -14 - 2i. Therefore, the answer is A. -14 - 2i.
2. To find the product of two complex numbers, you multiply them as you would with any other numbers. In this case, we have 8i * 5i. Remember that i squared is equal to -1. So, we can rewrite this expression as 8 * -1 * 5 * -1 = 40. Therefore, the answer is A. 40.
3. The product (5 + i)(5 - i) can be expanded using the distributive property. (5 + i)(5 - i) = 5 * 5 + 5 * (-i) + i * 5 + i * (-i) = 25 - 5i + 5i - i^2 = 25 - i^2. Since i squared is equal to -1, we can replace i^2 with -1. So, 25 -(-1) = 26. Thus, the factors (5 + i) and (5 - i) are called complex conjugates. Therefore, the answer is B. complex conjugates.
4. The product of complex conjugates is a difference of two squares and is always a real number. When you have a complex conjugate pair, you can multiply them together by using the formula (a + bi)(a - bi) = a^2 - (bi)^2. The imaginary terms will cancel each other out, leaving only the real terms. Therefore, the answer is C. The product of complex conjugates is a difference of two squares and is always a real number.
5. The expression a^6 + b^6 can be factored using the sum and difference of cubes formulas. The formula for the sum of cubes is a^3 + b^3 = (a + b)(a^2 - ab + b^2), and the formula for the difference of cubes is a^3 - b^3 = (a - b)(a^2 + ab + b^2). In this case, we have a^6 + b^6, which can be written as (a^3)^2 + (b^3)^2. Now, we can use the sum of squares formula, (a^2 + b^2)^2 = (a^2 - 2ab + b^2)(a^2 + 2ab + b^2). Replacing a^2 with a^3 and b^2 with b^3, we get (a^3 - 2ab^3 + b^3)(a^3 + 2ab^3 + b^3). Therefore, the answer is D. (a^3 - ib^2)(a^3 + ib^2).