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The given problem asks you to multiply two complex numbers and express the result in terms of \(i\). The expression is:

\[
(10 + 7i) \cdot (10 + 7i)
\]

This is equivalent to squaring the complex number \(10 + 7i\):

\[
(10 + 7i)^2
\]

To expand this, we use the distributive property (FOIL):

\[
(10 + 7i)(10 + 7i) = 10^2 + 2(10)(7i) + (7i)^2
\]

Now, calculating each term:

- \(10^2 = 100\)
- \(2(10)(7i) = 140i\)
- \((7i)^2 = 49i^2\)

Recall that \(i^2 = -1\), so:

\[
49i^2 = 49(-1) = -49
\]

Now, combine all the terms:

\[
100 + 140i - 49 = (100 - 49) + 140i = 51 + 140i
\]

Thus, the final simplified result is:

\[
51 + 140i
\]

Would you like further explanation on this? Here are a few questions related to complex number multiplication that might help you:

1. What is the general formula for multiplying two complex numbers?
2. How does the imaginary unit \(i\) behave under multiplication?
3. How does \(i^2 = -1\) affect complex number operations?
4. Can complex numbers be visualized geometrically in the complex plane?
5. How do powers of \(i\) cycle through different values?

**Tip:** When multiplying complex numbers, always apply the distributive property and remember that \(i^2 = -1\).

Learn how to multiply and simplify the complex numbers (10 + 7i) · (10 + 7i) using the FOIL method and properties of imaginary numbers. Includes step-by-step solution and explanation. Suitable for high school students.

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