Problem 6 - Precalculus: Mathematical for Calculus

Question

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

Step-by-Step Solution

Answer

The expanded form is \(a^3 + 6a^{5/2}b^{1/2} + 15a^2b + 20a^{3/2}b^{3/2} + 15ab^2 + 6a^{1/2}b^{5/2} + b^3\).

Step 1: Identify the row in Pascal's Triangle

Pascal's Triangle provides the binomial coefficients for expansions. For \((\sqrt{a} + \sqrt{b})^{6}\), we use the 7th row in Pascal's Triangle, which corresponds to \(n=6\). This row is: 1, 6, 15, 20, 15, 6, 1.

Step 2: Write the General Binomial Expansion Formula

The binomial expansion of \((x + y)^{n}\) is given by \( \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^{k} \). For our specific case, this becomes \( \sum_{k=0}^{6} \binom{6}{k} (\sqrt{a})^{6-k} (\sqrt{b})^{k} \).

Step 3: Apply the Binomial Coefficients for Each Term

Using the coefficients 1, 6, 15, 20, 15, 6, 1 from the 7th row of Pascal's Triangle, write the terms:1. \((\sqrt{a})^6 (\sqrt{b})^0 \rightarrow 1 \cdot (\sqrt{a})^6\)2. \(6(\sqrt{a})^5 (\sqrt{b})^1 \rightarrow 6 \cdot (\sqrt{a})^5 \cdot \sqrt{b}\)3. \(15(\sqrt{a})^4 (\sqrt{b})^2 \rightarrow 15 \cdot (\sqrt{a})^4 \cdot (\sqrt{b})^2\)4. \(20(\sqrt{a})^3 (\sqrt{b})^3 \rightarrow 20 \cdot (\sqrt{a})^3 \cdot (\sqrt{b})^3\)5. \(15(\sqrt{a})^2 (\sqrt{b})^4 \rightarrow 15 \cdot (\sqrt{a})^2 \cdot (\sqrt{b})^4\)6. \(6(\sqrt{a})^1 (\sqrt{b})^5 \rightarrow 6 \cdot \sqrt{a} \cdot (\sqrt{b})^5\)7. \((\sqrt{a})^0 (\sqrt{b})^6 \rightarrow 1 \cdot (\sqrt{b})^6\)

Step 4: Simplify Each Term

Now, calculate the powers of \(\sqrt{a}\) and \(\sqrt{b}\):1. \( (\sqrt{a})^6 = a^3 \)2. \( (\sqrt{a})^5 \cdot \sqrt{b} = a^{5/2} \cdot b^{1/2} \)3. \( (\sqrt{a})^4 \cdot (\sqrt{b})^2 = a^2 \cdot b \)4. \( (\sqrt{a})^3 \cdot (\sqrt{b})^3 = a^{3/2} \cdot b^{3/2} \)5. \( (\sqrt{a})^2 \cdot (\sqrt{b})^4 = a \cdot b^2 \)6. \( \sqrt{a} \cdot (\sqrt{b})^5 = a^{1/2} \cdot b^{5/2} \)7. \( (\sqrt{b})^6 = b^3 \)

Step 5: Combine the Terms to Form the Expansion

Combine all the simplified terms from Step 4 into the expanded binomial form:\[ a^3 + 6a^{5/2}b^{1/2} + 15a^2b + 20a^{3/2}b^{3/2} + 15ab^2 + 6a^{1/2}b^{5/2} + b^3 \]

Key Concepts

Binomial Expansion

The binomial expansion is an essential mathematical tool used to express the powers of binomials as a sum involving terms of a specific degree. In the case of the binomial expression \((x + y)^n\), the expansion results in a series of terms each of which has a specific coefficient known as a binomial coefficient. The expansion takes the form: \[ \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^{k} \] where \(\binom{n}{k}\) are the binomial coefficients derived from Pascal's Triangle. As a visual aid, Pascal’s Triangle helps in finding these coefficients easily. Each row of Pascal’s Triangle correlates to the degree \(n\), offering the needed coefficients. When expanding, each term involves power combinations of \(x\) and \(y\), distributed in decreasing and increasing orders respectively. With our expansion, \((\sqrt{a} + \sqrt{b})^6\), we're utilizing a sixth-degree binomial, and expanding it will give us 7 terms, facilitated by the structure of Pascal’s Triangle.

Binomial Coefficients

These coefficients are pivotal in determining the weight of each term in the binomial expansion. They are derived from Pascal's Triangle, where each element represents the value of \(\binom{n}{k}\) (read as "n choose k"). This notation describes the number of ways to choose \(k\) elements from \(n\) elements, a concept closely linked to combinatorics. For example, the coefficients for a sixth power are the numbers in the seventh row: 1, 6, 15, 20, 15, 6, 1.
These coefficients coincide with the arrangement of terms in the expression. As we prepare to expand \((\sqrt{a} + \sqrt{b})^6\), these numbers help determine how many times each individual term contributes to the sum. Specifically, each term takes the form \(\binom{6}{k} (\sqrt{a})^{6-k} (\sqrt{b})^k\), where \(k\) varies from 0 to 6. Understanding binomial coefficients ensures that each term follows the pattern dictated by Pascal's Triangle, accurately depicting the nature of the expanded expression.

Simplifying Expressions

Simplifying expressions is the step where the powers of the terms are calculated and combined appropriately to get the final expanded form. When dealing with expressions like \((\sqrt{a})^n\) or \((\sqrt{b})^n\), one should recognize that multiplying these radical terms results in powers that convert to integer values. For example, \((\sqrt{a})^2 = a\) and similarly \((\sqrt{a})^6 = a^3\).
The process involves evaluating these powers across each term in the expanded expression. During the expansion of \((\sqrt{a} + \sqrt{b})^6\), each term is simplified by first calculating the power of \(\sqrt{a}\) and \(\sqrt{b}\), followed by multiplying with the binomial coefficient obtained from Pascal's Triangle.

• For instance, \((\sqrt{a})^5 \cdot \sqrt{b} = a^{5/2} \cdot b^{1/2}\).
• Similarly, combining \((\sqrt{a})^3\) and \((\sqrt{b})^3\) results in \(a^{3/2} \cdot b^{3/2}\).
• Each term is adjusted and combined to represent simplified powers of \(a\) and \(b\).

By properly simplifying each element of the expansion, the expression \((\sqrt{a} + \sqrt{b})^6\) is fully expanded and simplified into its component parts, forming a cohesive mathematical result.