To solve the expression \(1 - x + x^2 - x^3 + \cdots + x^8\) where \(x = \sqrt{2}\), we first substitute \(x = \sqrt{2}\) into the expression:

\[
1 - \sqrt{2} + (\sqrt{2})^2 - (\sqrt{2})^3 + (\sqrt{2})^4 - (\sqrt{2})^5 + (\sqrt{2})^6 - (\sqrt{2})^7 + (\sqrt{2})^8
\]

Now, let's calculate each power of \(\sqrt{2}\):

\[
\begin{aligned}
(\sqrt{2})^2 & = 2, \\
(\sqrt{2})^3 & = 2\sqrt{2}, \\
(\sqrt{2})^4 & = 4, \\
(\sqrt{2})^5 & = 4\sqrt{2}, \\
(\sqrt{2})^6 & = 8, \\
(\sqrt{2})^7 & = 8\sqrt{2}, \\
(\sqrt{2})^8 & = 16.
\end{aligned}
\]

Substituting these values back into the original expression:

\[
1 - \sqrt{2} + 2 - 2\sqrt{2} + 4 - 4\sqrt{2} + 8 - 8\sqrt{2} + 16
\]

Next, let's combine like terms:

- Constant terms: \(1 + 2 + 4 + 8 + 16 = 31\)
- Terms with \(\sqrt{2}\): \(-\sqrt{2} - 2\sqrt{2} - 4\sqrt{2} - 8\sqrt{2} = -15\sqrt{2}\)

So, the expression simplifies to:

\[
31 - 15\sqrt{2}
\]

This is the simplified form of the expression when \(x = \sqrt{2}\).

Would you like more details or have any other questions? Here are some related questions you might find interesting:

1. How would the expression change if the sign pattern were reversed?
2. What happens to the expression if \(x\) is a complex number?
3. How can the same approach be applied if the series was infinite?
4. Can this be expressed in a geometric series form?
5. How does the value change if you extend the series to \(x^{10}\)?

**Tip:** When working with exponents involving square roots, it's helpful to recognize patterns, as they can simplify calculations significantly.

Learn how to evaluate the series 1 - x + x^2 - x^3 + ... + x^8 with x = √2. Step-by-step solution provided, including substitution of values and simplification. Suitable for students in Grades 10-12 studying series and exponents.

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