((1+i)^8=〖(√2)〗^8 (cos⁡〖8x π/4〗+sin⁡〖8x π/4〗))

We are given the equation:

\[
(1 + i)^8 = (\sqrt{2})^8 (\cos(8x \frac{\pi}{4}) + i \sin(8x \frac{\pi}{4}))
\]

Let's solve this step by step:

### Step 1: Simplifying \((1 + i)^8\)

We will convert \(1 + i\) into polar form. The complex number \(1 + i\) can be written as:

\[
1 + i = \sqrt{1^2 + 1^2} \left( \cos\left(\frac{\pi}{4}\right) + i \sin\left(\frac{\pi}{4}\right) \right)
\]

Here, \(\sqrt{1^2 + 1^2} = \sqrt{2}\), and the argument is \(\frac{\pi}{4}\). Therefore:

\[
1 + i = \sqrt{2} \left( \cos\left(\frac{\pi}{4}\right) + i \sin\left(\frac{\pi}{4}\right) \right)
\]

Now, raise both sides to the power of 8:

\[
(1 + i)^8 = (\sqrt{2})^8 \left( \cos\left(8 \times \frac{\pi}{4}\right) + i \sin\left(8 \times \frac{\pi}{4}\right) \right)
\]

\[
(1 + i)^8 = 16 \left( \cos\left(2\pi\right) + i \sin\left(2\pi\right) \right)
\]

Since \(\cos(2\pi) = 1\) and \(\sin(2\pi) = 0\), we get:

\[
(1 + i)^8 = 16
\]

### Step 2: Simplifying the right-hand side

The right-hand side of the equation is:

\[
(\sqrt{2})^8 \left( \cos(8x \frac{\pi}{4}) + i \sin(8x \frac{\pi}{4}) \right)
\]

Since \((\sqrt{2})^8 = 16\), we have:

\[
16 \left( \cos(8x \frac{\pi}{4}) + i \sin(8x \frac{\pi}{4}) \right)
\]

### Step 3: Equating both sides

Now we equate the two expressions:

\[
16 = 16 \left( \cos(8x \frac{\pi}{4}) + i \sin(8x \frac{\pi}{4}) \right)
\]

For this equation to hold, the imaginary part must be zero, and the real part must be 1. Therefore:

\[
\sin(8x \frac{\pi}{4}) = 0 \quad \text{and} \quad \cos(8x \frac{\pi}{4}) = 1
\]

### Step 4: Solving for \(x\)

The sine function is zero when its argument is an integer multiple of \(\pi\):

\[
8x \frac{\pi}{4} = n\pi \quad \Rightarrow \quad 2x\pi = n\pi \quad \Rightarrow \quad x = \frac{n}{2}
\]

The cosine function is 1 when its argument is an even multiple of \(2\pi\):

\[
8x \frac{\pi}{4} = 2k\pi \quad \Rightarrow \quad 2x = 2k \quad \Rightarrow \quad x = k
\]

Thus, \(x = k\), where \(k\) is an integer.

### Final Answer:
The value of \(x\) is an integer. 

Would you like more details or have any questions?

Here are 5 related questions for you to explore:

1. What is the polar form of other complex numbers like \(1 - i\)?
2. How do you find the roots of a complex number in polar form?
3. What is the geometric interpretation of complex numbers on the Argand plane?
4. How does De Moivre's theorem help in raising complex numbers to a power?
5. What are the properties of trigonometric functions that help solve complex equations?

**Tip:** When dealing with complex numbers, converting them into polar form can simplify multiplication, division, and exponentiation tasks.

(1+i)^8 = (√2)^8 (cos(8x π/4) + sin(8x π/4))

Explore how to solve the equation (1+i)^8 = (√2)^8 (cos(8x π/4) + sin(8x π/4)) using polar form and De Moivre's Theorem. This solution involves complex numbers, trigonometric functions, and exponentiation techniques. Suitable for undergraduate students studying complex numbers and polar coordinates.

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