To solve $(7 + 7i)^2$ and express the answer in the form $a + bi$:

Step 1: Expand using the square formula

Using the formula $(a + b)^2 = a^2 + 2ab + b^2$, let $a = 7$ and $b = 7i$:

$(7 + 7i)^2 = 7^2 + 2 \cdot 7 \cdot 7i + (7i)^2$

Step 2: Calculate each term

1. First term: $7^2 = 49$
2. Second term: $2 \cdot 7 \cdot 7i = 98i$
3. Third term: $(7i)^2 = 49i^2$

Since $i^2 = -1$, we have:

$49i^2 = 49 \cdot (-1) = -49$

Step 3: Combine terms

Substitute back:

$(7 + 7i)^2 = 49 + 98i - 49$

Now, combine the real parts:

$49 - 49 + 98i = 0 + 98i$

So, the answer in the form $a + bi$ is:

$0 + 98i \quad \text{or simply} \quad 98i$

Final Answer

$(7 + 7i)^2 = 98i$

Would you like more details or have any questions?

Related Questions:

1. How do you multiply two complex numbers?
2. What is the significance of the imaginary unit $i$ in complex numbers?
3. How do you expand $(a + bi)^n$ for any integer $n$?
4. What are the steps to convert a complex number to polar form?
5. How does squaring affect the magnitude and angle of a complex number?

Tip

When squaring a complex number, always remember to use $i^2 = -1$ to simplify the expression.