$$ \left(x^{2^{k}}-1\right)\left(x^{2^{k}}+1\right) $$

The given expression is:

\[
\left(x^{2^{k}} - 1\right)\left(x^{2^{k}} + 1\right)
\]

To simplify this, let's recognize that it follows the form $(a - b)(a + b) = a^2 - b^2$.

In this case:

- $a = x^{2^k}$
- $b = 1$

Applying the formula, we get:

\[
\left(x^{2^{k}} - 1\right)\left(x^{2^{k}} + 1\right) = \left(x^{2^k}\right)^2 - 1^2
\]

Simplifying further:

\[
\left(x^{2^k}\right)^2 - 1 = x^{2^{k+1}} - 1
\]

So, the simplified form of the given expression is:

\[
\left(x^{2^{k}} - 1\right)\left(x^{2^{k}} + 1\right) = x^{2^{k+1}} - 1
\]

Would you like more details or have any questions on this?

Here are 5 related questions that may expand your understanding:

1. How would you simplify $\left(x^{2^{k+1}} - 1\right)\left(x^{2^{k+1}} + 1\right)$?
2. What is the result when you factor $x^{2^{n}} - 1$ for a general $n$?
3. How do you apply the difference of squares in other algebraic expressions?
4. Can you generalize the simplification for any $n$ in $\left(x^{2^{n}} - 1\right)(x^{2^{n}} + 1)$?
5. What are the roots of $x^{2^{k+1}} - 1$?

**Tip:** The difference of squares formula $(a^2 - b^2 = (a - b)(a + b))$ is a powerful tool for factoring polynomials.

Explore how to simplify the expression $(x^{2^{k}} - 1)(x^{2^{k}} + 1)$ by applying the difference of squares. Learn key algebraic concepts and step-by-step solutions, ideal for students in Grades 10-12.

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