91Ó°ÊÓ

The Binomial Theorem is a powerful tool in algebra. It allows you to easily expand expressions that are raised to a power. The general form of the Binomial Theorem for any positive integer n is:\[(a + b)^n = \binom{n}{0} a^n b^0 + \binom{n}{1} a^{n-1} b^1 + \binom{n}{2} a^{n-2} b^2 + ... + \binom{n}{n} a^0 b^n\]In simpler terms, the theorem provides a formula to expand any binomial expression. 'Binomial' simply means 'two terms'. When n = 2, this simplifies to our basic binomial expansion: \[(a + b)^2 = a^2 + 2ab + b^2\]This form is useful because it makes expanding something like \(\big(e^x + e^{-x}\big)^2\) straightforward. We substitute \(a\) with \(e^x\) and \(b\) with \(e^{-x}\).

Exponentiation involves raising a number (the base) to the power of an exponent. In the expression \(e^x\), 'e' is the base and 'x' is the exponent. Here are some key points about exponentiation:

• An exponent tells you how many times to multiply the base by itself.
• The expression \(e^x\) means 'e' multiplied by itself 'x' times.
• There are special rules for exponents, such as \(e^a \times e^b = e^{a+b}\).

In our exercise, we also encounter \(e^{-x}\). The negative exponent means reciprocal, so \(e^{-x} = \frac{1}{e^x}\). Understanding these rules will help simplify expressions involving exponents, such as \(\big(e^x + e^{-x}\big)^2\).

Simplifying expressions means making them as compact and readable as possible. This involves combining like terms and applying algebraic rules. For example:

• Combining like terms: When you expand \(\big(e^x + e^{-x}\big)^2\), you get \((e^x)^2 + 2(e^x)(e^{-x}) + (e^{-x})^2\).
• Simplifying each term:
  - \((e^x)^2 = e^{2x}\),
  - \(2(e^x)(e^{-x}) = 2e^0 = 2\) because any number raised to the power of zero is 1,
  - \((e^{-x})^2 = e^{-2x}\).
• Combining and writing the final simplified expression: \(e^{2x} + 2 + e^{-2x}\).

The key goal is to end up with a simpler version of the original expression while still maintaining equality.