Question

Вычислить√(-48+55i)​

Answer 1

Ответ:

$\sqrt{3 }$

Answer 2

To calculate the square root of a complex number, √(a + bi), where 'a' and 'b' are real numbers and 'i' is the imaginary unit (√(-1)), you can follow these steps:

Step 1: Write the complex number in the form a + bi. Step 2: Find the magnitude (or modulus) of the complex number, which is given by |z| = √(a^2 + b^2). Step 3: Find the argument (or angle) of the complex number, which is given by θ = atan(b/a). Step 4: The square root of the complex number is given by √(a + bi) = ±√(|z|) * [cos(θ/2) + i * sin(θ/2)].

Let's apply these steps to √(-48 + 55i):

Step 1: The complex number is -48 + 55i.

Step 2: Calculate the magnitude |z| = √((-48)^2 + 55^2) = √(2304 + 3025) = √5329 = 73.

Step 3: Calculate the argument θ = atan(55 / -48) ≈ -47.13 degrees (approximately).

Step 4: The square root of the complex number is √(-48 + 55i) = ±√(73) * [cos(-47.13/2) + i * sin(-47.13/2)].

Now, we need to compute cos(-47.13/2) and sin(-47.13/2):

cos(-47.13/2) ≈ 0.874, and sin(-47.13/2) ≈ -0.486.

Finally, the square root of -48 + 55i is:

√(-48 + 55i) ≈ ±√(73) * (0.874 - 0.486i).

So, the two possible square roots are approximately:

1. √(-48 + 55i) ≈ √(73) * (0.874 - 0.486i)
2. -√(-48 + 55i) ≈ -√(73) * (0.874 - 0.486i)

Note that the "±" sign indicates that there are two square roots for any complex number.