Find square root of th complex number 7 +24i ?

Question

Answer

Let the square root of 7 + 24i = a +bi where a and b are real numbers.

We know that i^2 = -1

Now (a+bi)^2 = 7 + 24i

=> a^2 -b^2 + 2(a)(b)i =7 + 24i

Comparing both sides we get two equations a^2 - b^2 =7

And 2ab = 24

=> ab = 12

=> b =12/a

Now putting the value of b in equation 1 we get

a^2 - (12/a)^2 = 7

=> a^2 -(144/a^2) = 7

=> a^4 -144 = 7a^2

=> a^4 - 7 a^2 -144 = 0

Solving above quadratic equation we get

a^2 = (7+√(576+49))/2 or a^2 = (7-√(576+49))/2

=> a^2 = (7+25)/2 or a^2= (7–25)/2

=> a^2 = 16 or a^2 = -9

Since we have assumed a as real a^2 can't be negative

Therefore a^2 = 16

=> a = 4 or a = -4

Putting the value of a in ab = 12 we get

b = 3 or b = -3

Therefore our required solution is 4 + 3i or -4 -3i

Answer

Answer

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.

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