Express the following in the form a + bi, where a and b are real numbers:
\sqrt{55+48i}
√(55+48i) = √(64 + 48i - 9) = √(55+48i) = √(8+3i)^2 = 8+3i
Well, let's start by simplifying the expression inside the square root:
55 + 48i = (a + bi)^2
Expanding the square:
55 + 48i = a^2 + 2abi - b^2
Now, let's equate the real and imaginary parts:
Real part: 55 = a^2 - b^2
Imaginary part: 48 = 2ab
From the imaginary part equation, we can solve for a:
24 = ab
Now, we substitute this into the real part equation:
55 = a^2 - (24/a)^2
55 = a^2 - 24^2/a^2
55 = a^4 - 576
a^4 = 631
a ≈ ±8.515
Now, let's find b:
24 = (8.515)b
b ≈ 24/8.515
b ≈ 2.82
So, expressing \sqrt{55+48i} in the form a + bi, where a and b are real numbers, we have approximately:
\sqrt{55+48i} ≈ 8.515 + 2.82i
To express \sqrt{55 + 48i} in the form a + bi, where a and b are real numbers, we need to find the square root of the given complex number.
Let's assume that \sqrt{55 + 48i} = a + bi, where a and b are real numbers.
Square both sides to get:
(55 + 48i) = (a + bi)^2
Expanding the right side using the binomial formula:
55 + 48i = a^2 + 2abi - b^2
Now let's equate the real and imaginary parts on both sides:
Real part:
55 = a^2 - b^2 ...(1)
Imaginary part:
48 = 2ab ...(2)
Now, we have a system of equations. Let's solve it.
From equation (2), we can rearrange it to find the value of ab:
ab = 24 ...(3)
By manipulating equation (1), we can solve for a^2:
a^2 = 55 + b^2 ...(4)
Substitute the value of a^2 from equation (4) into equation (3):
(55 + b^2)b = 24
Expanding and rearranging,
55b + b^3 = 24
This equation is nonlinear and will require approximation or numerical methods to solve. Evaluating the equation, we find that b ≈ 0.439 and a ≈ ±7.086.
Therefore, \sqrt{55 + 48i} can be expressed in the form a + bi as:
\sqrt{55 + 48i} ≈ 7.086 + 0.439i
or
\sqrt{55 + 48i} ≈ -7.086 - 0.439i
To express the given expression \sqrt{55+48i} in the form a + bi, where a and b are real numbers, we need to find the square root of the complex number 55 + 48i.
Let's calculate step by step:
Step 1: Find the modulus (absolute value) of the complex number.
The modulus of a complex number a + bi is given by |a + bi| = sqrt(a^2 + b^2).
In our case, the modulus of 55 + 48i is:
|55 + 48i| = sqrt(55^2 + 48^2)
= sqrt(3025 + 2304)
= sqrt(5329)
= 73.
So, the modulus is 73.
Step 2: Find the argument (angle) of the complex number.
The argument of a complex number a + bi is given by arg(a + bi) = arctan(b/a).
In our case, the argument of 55 + 48i is:
arg(55 + 48i) = arctan(48/55).
Using a calculator, we find:
arg(55 + 48i) ≈ 42.86° (rounded to two decimal places).
Step 3: Express the complex number in polar form.
A complex number in polar form is given by z = r(cosθ + isinθ), where r is the modulus and θ is the argument.
In our case, the complex number 55 + 48i can be expressed as:
55 + 48i = 73(cos(42.86°) + isin(42.86°)).
Step 4: Express the complex number in rectangular form.
To express the complex number in the form a + bi, we can expand the expression in step 3 using trigonometric identities.
Using the identity cosθ = Re^(iθ), where Re represents the real part, and sinθ = Im^(iθ), where Im represents the imaginary part, we can rewrite the expression:
55 + 48i = 73(cos(42.86°) + isin(42.86°))
= 73(e^(i(42.86°)))
= 73(e^(i(π/180)(42.86)))
Next, we can use Euler's formula, e^(ix) = cos(x) + isin(x), to write:
55 + 48i = 73(e^(i(π/180)(42.86)))
= 73(cos((π/180)(42.86)) + isin((π/180)(42.86)))
Now, let's calculate:
55 + 48i = 73(cos(0.7477) + isin(0.7477))
= 73(0.7314 + 0.6816i)
= 53.4072 + 49.6872i.
Thus, the expression \sqrt{55 + 48i} can be written as:
\sqrt{55 + 48i} = 7.3114 + 6.9858i (rounded to four decimal places).