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**Finding the Square Root of 1 + i**

To find the square root of 1 + i, we can use the concept of complex numbers and the properties of square roots.

**Complex Numbers**

Complex numbers are numbers that consist of two parts: a real part and an imaginary part. They are expressed in the form a + bi, where 'a' is the real part and 'b' is the imaginary part. In this case, we have 1 + i, where 1 is the real part and i is the imaginary part.

**Properties of Square Roots**

When finding the square root of a complex number, we can use the following properties of square roots:

1. The square root of a complex number has two solutions.
2. If z^2 = w, then the square roots of w are ± z.

**Finding the Square Root of 1 + i**

Let's find the square root of 1 + i using the properties mentioned above.

Step 1: Express 1 + i in polar form.
To simplify the calculation, we can convert the complex number 1 + i into polar form. The polar form of a complex number is expressed as r(cosθ + isinθ), where r is the modulus (or absolute value) of the complex number and θ is the argument (or angle) of the complex number.

To find the modulus (r) and argument (θ) of 1 + i, we can use the following formulas:

r = √(a^2 + b^2)
θ = arctan(b/a)

In this case, a = 1 and b = 1. Therefore,

r = √(1^2 + 1^2) = √2
θ = arctan(1/1) = arctan(1) = π/4

So, 1 + i can be expressed in polar form as √2(cos(π/4) + isin(π/4)).

Step 2: Take the square root of the polar form.
Using the properties of square roots, we can now find the square root of √2(cos(π/4) + isin(π/4)).

Since the square root has two solutions, we can express it as ±√2^(1/2)(cos(π/8) + isin(π/8)).

Simplifying further, we get:

±(2^(1/4))(cos(π/8) + isin(π/8))

So, the square root of 1 + i is ±(2^(1/4))(cos(π/8) + isin(π/8)).

In summary, the square root of 1 + i can be expressed as ±(2^(1/4))(cos(π/8) + isin(π/8)).