Complex And Imaginary Numbers Quiz

Explanation
The expression -3i(4 + 2i) can be simplified by distributing the -3i to both terms inside the parentheses. This results in -12i - 6i^2. Since i^2 is equal to -1, the expression simplifies further to -12i - 6(-1), which becomes -12i + 6. Finally, combining like terms, the simplified expression is 6 - 12i.

Explanation
The expression √-25 can be simplified as 5i. This is because the square root of -25 is equal to the square root of 25 multiplied by the square root of -1. The square root of 25 is 5, and the square root of -1 is represented by the imaginary unit i. Therefore, the simplified form of √-25 is 5i.

Explanation
The given question asks for the sum of two complex numbers, (5-2i) and (-7+8i). To find the sum, we simply add the real parts and the imaginary parts separately. Adding the real parts, 5 and -7, we get -2. Adding the imaginary parts, -2i and 8i, we get 6i. Therefore, the sum of (5-2i) and (-7+8i) is -2+6i.

Explanation
The solution to -4i + 7i can be found by combining the real and imaginary parts separately. The real part of -4i is 0, and the real part of 7i is also 0. Therefore, the real part of the sum is 0. The imaginary part of -4i is -4, and the imaginary part of 7i is 7. Therefore, the imaginary part of the sum is 7 - 4 = 3. Thus, the correct answer is 3i.

Explanation
To simplify the expression (-3-i)(6-i), we can use the FOIL method. First, we multiply the first terms: -3 * 6 = -18. Then, we multiply the outer terms: -3 * -i = 3i. Next, we multiply the inner terms: -i * 6 = -6i. Finally, we multiply the last terms: -i * -i = i^2 = -1. Combining all these results, we get -18 + 3i - 6i - 1. Simplifying further, we have -19 - 3i.

Explanation
To solve the given multiplication, we can use the FOIL method. Multiplying the first terms of each binomial gives us (4)(-7) = -28. Multiplying the outer terms gives us (4)(-2i) = -8i. Multiplying the inner terms gives us (-3i)(-7) = 21i. Multiplying the last terms gives us (-3i)(-2i) = 6i^2. Simplifying, we have -28 - 8i + 21i + 6i^2. Since i^2 is equal to -1, we can substitute it in the expression to get -28 - 8i + 21i + 6(-1). Combining like terms, we have -28 + 13i - 6, which simplifies to -34 + 13i. Therefore, the correct answer is -34 + 13i.

Explanation
The square root of -100 is equal to 10i. In complex numbers, the imaginary unit i is defined as the square root of -1. Therefore, the square root of -100 can be written as 10 * sqrt(-1), which simplifies to 10i.

Explanation
The square root of a negative number is an imaginary number. In this case, the square root of -81 is 9i, where i represents the imaginary unit.

Explanation
The product of (1+3i)(2-i) can be found by using the distributive property. First, multiply 1 by 2 and get 2. Then, multiply 1 by -i and get -i. Next, multiply 3i by 2 and get 6i. Finally, multiply 3i by -i and get -3i^2. Simplifying further, -3i^2 can be written as -3(-1), which is equal to 3. Combining all the terms, we get 2 + (-i) + 6i + 3, which simplifies to 5 + 5i. Therefore, the correct answer is 5 + 5i.

Explanation
The basic format for writing a complex number is a+bi, where 'a' represents the real part of the number and 'b' represents the imaginary part. The imaginary part is multiplied by 'i', the imaginary unit. This format allows us to represent numbers that have both a real and imaginary component, such as 3+2i, where 3 is the real part and 2i is the imaginary part. The other options provided, a-bi and ai+b, do not follow the standard format for writing complex numbers.