Problem 12 - Precalculus Mathematics for Calculus

Question

Use Pascal's triangle to expand the expression. $$(1+\sqrt{2})^{6}$$

Step-by-Step Solution

Answer

\((1+\sqrt{2})^6 = 99 + 70\sqrt{2}\).

Step 1: Understand Pascal's Triangle

Pascal's Triangle is a triangular array of binomial coefficients. Each row corresponds to the coefficients in the expansion of a binomial expression of increasing powers such as \((a + b)^n\). For example, the 6th row (starting from 0) will contain the coefficients for \((a + b)^6\). We need to identify this row from Pascal's Triangle.

Step 2: Identify the Correct Row in Pascal's Triangle

Pascal's Triangle starts with 1 at the top. For \((1 + \sqrt{2})^6\), we need the 7th row (since it starts from a \((a + b)^0\) row). The row for \((a + b)^6\) is: 1, 6, 15, 20, 15, 6, 1.

Step 3: Apply Binomial Expansion Formula

Using the binomial expansion formula for \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k}b^k\), where \(\binom{n}{k}\) are the binomial coefficients. Let's apply this for \((1 + \sqrt{2})^6\) with our coefficients from Pascal's Triangle.

Step 4: Calculate Each Term in the Expansion

Replace \(a\) with 1 and \(b\) with \(\sqrt{2}\). Using the coefficients from Pascal's Triangle (1, 6, 15, 20, 15, 6, 1) and applying each power of \(a\) and \(b\):- First term: \(1 \cdot (1)^6 \cdot (\sqrt{2})^0 = 1\)- Second term: \(6 \cdot (1)^5 \cdot (\sqrt{2})^1 = 6\sqrt{2}\)- Third term: \(15 \cdot (1)^4 \cdot (\sqrt{2})^2 = 30\)- Fourth term: \(20 \cdot (1)^3 \cdot (\sqrt{2})^3 = 40\sqrt{2}\)- Fifth term: \(15 \cdot (1)^2 \cdot (\sqrt{2})^4 = 60\)- Sixth term: \(6 \cdot (1)^1 \cdot (\sqrt{2})^5 = 24\sqrt{2}\)- Seventh term: \(1 \cdot (1)^0 \cdot (\sqrt{2})^6 = 8\)

Step 5: Sum All Terms

Combine all the terms from the expansion:1. Coefficients: 1, 30, 60, 82. Terms with square root: 6\sqrt{2}, 40\sqrt{2}, 24\sqrt{2}Sum them to get:- Coefficients: \(99\)- Square root terms: \(70\sqrt{2}\)So, \((1 + \sqrt{2})^6 = 99 + 70\sqrt{2}\).

Key Concepts

Binomial Expansion

The binomial expansion is a mathematical technique used to expand expressions that are to a power, typically in the form of \((a + b)^n\). This formula allows us to break down a more complex algebraic expression into a series of simpler terms.

Essentially, each term in the expansion is found by using the binomial coefficients, which are derived from Pascal's Triangle, a tool we'll talk about in the next section. Here's how the binomial expansion is structured:

\[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k}b^k\]

• \(n\) is the exponent to which the binomial is raised.
• \(k\) ranges from 0 up to \(n\), representing the term we are calculating.
• \(\binom{n}{k}\) is the binomial coefficient, which tells you how many ways you can choose \(k\) elements from \(n\).

This method gives us a systematic way to distribute powers across the terms inside the binomial expression. Understanding this will help you work through complex algebraic problems, like expanding \((1 + \sqrt{2})^6\).

In such expansions, the terms will alternate through the coefficients provided by Pascal's Triangle with corresponding powers of both binomial components \(a\) and \(b\). Use it to simplify binomial expressions efficiently!

Binomial Coefficients

Binomial coefficients are integral parts of binomial expansions, providing the numerical factors for each term within the expansion. These coefficients can be directly obtained from Pascal's Triangle, a triangular array of numbers.

Pascals Triangle is constructed such that:

• The first and last element of every row is 1.
• Every intermediate element is the sum of the two numbers directly above it in the previous row.

The coefficients determine the number of combinations possible for choosing elements, hence affecting the proportionate weight of each term in the expansion.

For example, in the expression \((1 + \sqrt{2})^6\), we use the 7th row (starting with \((a + b)^0\)) which is: \(1, 6, 15, 20, 15, 6, 1\).

Each binomial coefficient corresponds to a term in the expansion:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]

• \(\binom{n}{k}\) is the coefficient of the term containing \(a^{n-k}b^k\).
• These values give multiples to the product of powers of \(a\) and \(b\).

Knowing how to use binomial coefficients ensures you can calculate any term in a binomial expansion accurately. Understanding them is critical in mastering binomial expansions.

Exponents

Exponents are used to indicate that a number should be multiplied by itself a certain number of times. They are central to calculating the powers of terms within binomial expansions.

In an expression like \((a+b)^n\):

• \(n\) is the exponent, indicating how many times the whole binomial is used as a factor.
• The term \(a^{n-k}\) tells you how many times \(a\) is multiplied by itself, and \(b^k\) indicates the same for \(b\).

For instance, in \((1 + \sqrt{2})^6\), you must calculate various powers:

• \((1)^6, (1)^5, ..., (1)^0\)
• \((\sqrt{2})^0, (\sqrt{2})^1, ..., (\sqrt{2})^6\)

Powers of \(1\) simplify to \(1\) itself when \(n eq 0\). However, the powers of \(\sqrt{2}\) will yield different results due to squaring.

Each term's contribution in the expansion results from these exponent combinations. Understanding exponents lets you correctly simplify each term, ensuring accurate expansion of any binomial expression.