Problem 25 - Algebra and Trigonometry

Question

Expand each expression using the Binomial Theorem. $$ (\sqrt{x}+\sqrt{2})^{6} $$

Step-by-Step Solution

Answer

short answer

Step 1: Understand the Binomial Theorem

The Binomial Theorem states that expressing where

Step 2: Identify Each Part

In the expression and

Step 3: Write the General Term

Each term in the expansion is of the form

Step 4: Expand Each Term

Expand each term using the combination formula and simplify.

Step 5: Write the Full Expanded Form

Combine all the terms to write the full expanded form of the expression.

Key Concepts

Binomial Expansion

Binomial expansion is a powerful algebraic tool used to expand expressions of the form \( (a + b)^n \). Instead of having to multiply the binomials repeatedly, the Binomial Theorem provides a straightforward way to expand these expressions. For any positive integer \( n \), the binomial expansion is given by:
\[ (a+b)^n = \sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k \]
This formula tells us that each term in the expansion can be found using the combination formula, along with powers of \( a \) and \( b \). Understanding binomial expansion is essential for simplifying and working with polynomial expressions more efficiently.

Combination Formula

The combination formula is used to determine the coefficients in the terms of a binomial expansion. It is represented by \( \binom{n}{k} \), also known as 'n choose k' or binomial coefficients. The formula for combinations is:
\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]
Here, \( n! \) stands for factorial of \( n \), which is the product of all positive integers up to \( n \). The combination formula essentially calculates how many ways we can choose \( k \) elements from a set of \( n \) elements. In the context of binomial expansion, this coefficient helps in constructing each term in the expanded form.

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, etc. When expanding expressions with radicals using the Binomial Theorem, it’s important to handle the radicals accurately. For example, consider the expression:
\[ (\sqrt{x} + \sqrt{2})^6 \]
The variable \( a \) is \( \sqrt{x} \) and \( b \) is \( \sqrt{2} \). While expanding this expression using the binomial theorem, each term will include different powers of \( \sqrt{x} \) and \( \sqrt{2} \). It's important to simplify each term correctly, keeping in mind the properties of radicals, such as \( (\sqrt{x})^2 = x \).

Mathematical Notation

Mathematical notation makes it possible to write mathematical concepts in a concise and unambiguous way. In the Binomial Theorem, a few key pieces of notation are critical. They include:

• \(n!\) which denotes the factorial of \( n \)
• \(\binom{n}{k}\) which represents combinations or binomial coefficients
• \( (a + b)^n \) which is the general form of a binomial expression

Using proper notation helps in understanding and communicating mathematical ideas effectively. When studying binomial expansion, pay close attention to these notations to ensure that you interpret and solve the problems correctly.