To multiply \((t^a + t^{-a})^2\), we apply the distributive property (also known as the FOIL method when working with binomials). Here’s the step-by-step process:

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

Now, distribute each term in the first binomial to each term in the second binomial:

\[
t^a \cdot t^a + t^a \cdot t^{-a} + t^{-a} \cdot t^a + t^{-a} \cdot t^{-a}
\]

Now, simplify each term:

1. \( t^a \cdot t^a = t^{a+a} = t^{2a} \)
2. \( t^a \cdot t^{-a} = t^{a-a} = t^0 = 1 \)
3. \( t^{-a} \cdot t^a = t^{-a+a} = t^0 = 1 \)
4. \( t^{-a} \cdot t^{-a} = t^{-a-a} = t^{-2a} \)

Now, combine all the terms:

\[
t^{2a} + 1 + 1 + t^{-2a}
\]

Simplify further:

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

Thus, the result of multiplying \((t^a + t^{-a})^2\) is:

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

Would you like further details or explanations?

Here are some related questions:
1. How would this change if the exponent were different, such as \((t^a + t^{-a})^3\)?
2. Can you derive a general formula for \((t^a + t^{-a})^n\)?
3. What happens when \(a = 1\) in this expression?
4. How do you factor expressions like \(t^{2a} + 2 + t^{-2a}\)?
5. What are the real-world applications of simplifying binomials like this?

**Tip:** Remember that multiplying powers with the same base involves adding exponents.

Learn how to multiply (t^a + t^-a)^2 using the distributive property and exponent rules. This step-by-step guide simplifies the algebraic expression, making it perfect for students in Grades 9-11. Understand key concepts like binomial expansion and exponent rules.

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