To solve $(1 + i)^7$, you can expand it using the binomial theorem or use polar (or exponential) form for complex numbers. I'll explain both methods for you.

Method 1: Binomial Expansion

We can expand $(1 + i)^7$ using the binomial theorem:

$(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$

For $a = 1$, $b = i$, and $n = 7$, we expand $(1 + i)^7$:

$(1 + i)^7 = \sum_{k=0}^{7} \binom{7}{k} 1^{7-k} i^k$

This simplifies to:

$(1 + i)^7 = \sum_{k=0}^{7} \binom{7}{k} i^k$

Since the powers of $i$ cycle every 4 terms (i.e., $i^1 = i$, $i^2 = -1$, $i^3 = -i$, and $i^4 = 1$), you can substitute these values for the powers of $i$.

Expanding the sum:

$= \binom{7}{0} i^0 + \binom{7}{1} i^1 + \binom{7}{2} i^2 + \binom{7}{3} i^3 + \binom{7}{4} i^4 + \binom{7}{5} i^5 + \binom{7}{6} i^6 + \binom{7}{7} i^7$

Simplifying the individual terms:

$= 1 + 7i + 21(-1) + 35(-i) + 35(1) + 21i + 7(-1) + 1i$

Now, group real and imaginary parts:

Real part: $1 - 21 + 35 - 7 = 8$

Imaginary part: $7i - 35i + 21i + i = -6i$

Thus: $(1 + i)^7 = 8 - 6i$

Method 2: Using Polar Form

Alternatively, you can express $1 + i$ in polar form.

1. Find the modulus $r$ and argument $\theta$ of $1 + i$:

• Modulus: $r = |1 + i| = \sqrt{1^2 + 1^2} = \sqrt{2}$

• Argument $\theta$ (since $1 + i$ is in the first quadrant): $\theta = \tan^{-1}\left(\frac{1}{1}\right) = \frac{\pi}{4}$

So, $1 + i = \sqrt{2} e^{i\pi/4}$.

2. Now, apply De Moivre's Theorem for $(1 + i)^7$:

$(1 + i)^7 = \left( \sqrt{2} e^{i\pi/4} \right)^7 = (\sqrt{2})^7 e^{i7\pi/4}$

Simplifying the modulus: $(\sqrt{2})^7 = 2^{7/2} = 8\sqrt{2}$

And the argument: $e^{i7\pi/4} = e^{i(-\pi/4)} \quad (\text{since } 7\pi/4 \text{ is equivalent to } -\pi/4 \text{ mod } 2\pi)$

Thus, we have: $(1 + i)^7 = 8\sqrt{2} e^{-i\pi/4}$

Now, convert this back to rectangular form: $8\sqrt{2} \left( \cos\left( -\frac{\pi}{4} \right) + i \sin\left( -\frac{\pi}{4} \right) \right)$ $= 8\sqrt{2} \left( \frac{1}{\sqrt{2}} - i \frac{1}{\sqrt{2}} \right)$ $= 8(1 - i) = 8 - 8i$

Thus, we get the same result as earlier: $(1 + i)^7 = 8 - 8i$

Final Answer:

$(1 + i)^7 = 8 - 8i$

Would you like more details on either method, or have any questions about the process?

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Here are some related questions you might find interesting:

1. How do you convert complex numbers to polar form?
2. What is De Moivre's Theorem, and how is it used with complex numbers?
3. How do you simplify higher powers of complex numbers?
4. What is the binomial theorem, and how can it be used for expanding complex numbers?
5. How do you find the argument of a complex number geometrically?

Tip: When working with complex numbers, converting to polar form often simplifies raising numbers to higher powers or taking roots.