https://file.aiquickdraw.com/mathbot/file/17273739125483ntcawu2.jpg

The expression shown in the image is:

\[
(a - a\sqrt{2})^2
\]

We can simplify this by expanding the square using the binomial expansion formula:

\[
(x - y)^2 = x^2 - 2xy + y^2
\]

Here, \(x = a\) and \(y = a\sqrt{2}\). Applying the formula:

\[
(a - a\sqrt{2})^2 = a^2 - 2(a)(a\sqrt{2}) + (a\sqrt{2})^2
\]

Now simplify each term:

1. \(a^2\) stays as is.
2. \(-2(a)(a\sqrt{2}) = -2a^2\sqrt{2}\).
3. \((a\sqrt{2})^2 = a^2 \cdot 2 = 2a^2\).

Putting it all together:

\[
a^2 - 2a^2\sqrt{2} + 2a^2
\]

This is the fully simplified form:

\[
a^2 - 2a^2\sqrt{2} + 2a^2
\]

Let me know if you need any further details or clarifications!

Here are 5 related questions to explore:

1. How would the expansion change if the square root was \(\sqrt{3}\) instead of \(\sqrt{2}\)?
2. Can you apply this binomial expansion to a cubic expression like \((a - a\sqrt{2})^3\)?
3. What happens if you replace \(a\) with a specific value, like 1 or 2, and simplify further?
4. How would the result change if the sign in the original expression was positive, i.e., \((a + a\sqrt{2})^2\)?
5. How is the binomial expansion used in real-world applications, such as physics or engineering?

**Tip**: Binomial expansions become very helpful in algebra when dealing with powers of binomials. Mastering this technique will aid in solving more complex equations easily.

Learn how to simplify the algebraic expression (a - a√2)^2 using binomial expansion. This solution walks through the expansion formula step by step and explains each term's simplification. Suitable for students in Grades 9-12.

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