Introduction:

To find the square root of a complex number, we need to express it in the form a + bi, where a and b are real numbers. In this case, we have the complex number 7 + 24i.

Step 1: Separate the real and imaginary parts:

To find the square root, we need to separate the real and imaginary parts of the complex number. In this case, the real part is 7 and the imaginary part is 24i.

Step 2: Express the complex number in the form a + bi:

Since the complex number is already in the form a + bi, there is no need to make any changes.

Step 3: Find the magnitude of the complex number:

The magnitude of a complex number is given by the formula |z| = √(a^2 + b^2), where a and b are the real and imaginary parts of the complex number, respectively. In this case, the magnitude is |z| = √(7^2 + 24^2) = √(49 + 576) = √625 = 25.

Step 4: Find the principal argument of the complex number:

The principal argument of a complex number is the angle it forms with the positive real axis in the complex plane. It can be found using the formula tan^(-1)(b/a), where a and b are the real and imaginary parts of the complex number, respectively. In this case, the principal argument is tan^(-1)(24/7).

Step 5: Express the square root in the form a + bi:

To find the square root, we can use the formula √(r) * [cos(θ/2) + i * sin(θ/2)], where r is the magnitude of the complex number and θ is the principal argument.

Step 6: Calculate the square root:

Using the formula from step 5, we can calculate the square root of the complex number 7 + 24i as follows:

√(25) * [cos(tan^(-1)(24/7)/2) + i * sin(tan^(-1)(24/7)/2)]

Simplifying this expression will give us the square root of the complex number.

Conclusion:

The square root of the complex number 7 + 24i is obtained by separating the real and imaginary parts, finding the magnitude and principal argument, and then using the formula for the square root of a complex number.