Problem 10 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

\(a^{3} + 6a^{2.5}b^{0.5} + 15a^{2}b + 20a^{1.5}b^{1.5} + 15ab^{2} + 6a^{0.5}b^{2.5} + b^{3}\)

Step 1: Identify the Row of Pascal's Triangle

We need to expand \((\sqrt{a} + \sqrt{b})^6\). First, identify the 6th row of Pascal's Triangle, which gives us the coefficients for the expansion. The 6th row is: 1, 6, 15, 20, 15, 6, 1.

Step 2: Determine the Terms

Each term in the expansion follows the form: \( \binom{n}{k} \cdot (\sqrt{a})^{n-k} \cdot (\sqrt{b})^{k} \). For \((\sqrt{a} + \sqrt{b})^6\), each term will have exponent sums equal to 6.

Step 3: Write the Expanded Form

Using the coefficients from Pascal's Triangle and alternating powers of \(\sqrt{a}\) and \(\sqrt{b}\), write the expanded form:1. \(1 \cdot (\sqrt{a})^{6}\)2. \(6 \cdot (\sqrt{a})^{5} \cdot (\sqrt{b})^{1}\)3. \(15 \cdot (\sqrt{a})^{4} \cdot (\sqrt{b})^{2}\)4. \(20 \cdot (\sqrt{a})^{3} \cdot (\sqrt{b})^{3}\)5. \(15 \cdot (\sqrt{a})^{2} \cdot (\sqrt{b})^{4}\)6. \(6 \cdot (\sqrt{a})^{1} \cdot (\sqrt{b})^{5}\)7. \(1 \cdot (\sqrt{b})^{6}\)Let's write each term clearly in the expanded form.

Step 4: Calculate Each Term and Simplify

Now simplify the powers and calculate:1. \((\sqrt{a})^{6} = a^{3}\)2. \(6 \cdot (\sqrt{a})^{5} \cdot (\sqrt{b})^{1} = 6a^{2.5}b^{0.5}\)3. \(15 \cdot (\sqrt{a})^{4} \cdot (\sqrt{b})^{2} = 15a^{2}b\)4. \(20 \cdot (\sqrt{a})^{3} \cdot (\sqrt{b})^{3} = 20a^{1.5}b^{1.5}\)5. \(15 \cdot (\sqrt{a})^{2} \cdot (\sqrt{b})^{4} = 15ab^{2}\)6. \(6 \cdot (\sqrt{a})^{1} \cdot (\sqrt{b})^{5} = 6a^{0.5}b^{2.5}\)7. \((\sqrt{b})^{6} = b^{3}\)

Step 5: Combine to Get the Final Expansion

Combine all the terms together:\[a^{3} + 6a^{2.5}b^{0.5} + 15a^{2}b + 20a^{1.5}b^{1.5} + 15ab^{2} + 6a^{0.5}b^{2.5} + b^{3}\]

Key Concepts

Binomial Expansion

Binomial expansion is a method used to expand expressions of the form \((x + y)^n\), where \(x\) and \(y\) are any expressions and \(n\) is a positive integer. The coefficients of the expansion are given by the binomial coefficients, which can be found using Pascal's Triangle. Each coefficient in Pascal’s Triangle corresponds to a term in the binomial expansion.For example, when expanding \((\sqrt{a} + \sqrt{b})^6\), we obtain:

• First, identify the 6th row of Pascal's triangle: 1, 6, 15, 20, 15, 6, 1.
• Each term in the expansion takes the form \(\binom{n}{k} \cdot (\text{first term})^{n-k} \cdot (\text{second term})^{k}\). In our example, the first term is \(\sqrt{a}\) and the second term is \(\sqrt{b}\).

The binomial expansion is not only used for simple powers. It has applications in probability, algebra, and calculus, where it helps in series expansions and solving equations. By learning binomial expansion, you can develop strategies for dealing with complex algebraic expressions.

Powers and Exponents

Powers and exponents are fundamental concepts in mathematics that describe how many times a number, known as the base, is multiplied by itself. In the expression \((\sqrt{a} + \sqrt{b})^6\), both \(\sqrt{a}\) and \(\sqrt{b}\) are bases raised to different powers throughout the expansion process.Understanding powers and exponents allows you to simplify expressions and solve equations more effectively:

• If a number is raised to an exponent, for instance, \(a^3\), it means \(a\) is used as a factor three times: \(a \cdot a \cdot a\).
• When raising a power to another power, you multiply the exponents: \((a^m)^n = a^{m \cdot n}\).
• For fractional exponents, like \(a^{1/2}\), the base is subject to a root: \(a^{1/2} = \sqrt{a}\).

Powers and exponents significantly ease the process of dealing with polynomial expressions, like our example's terms. By practicing manipulating exponents, you improve your ability to handle even complex operations in algebra.

Algebraic Expressions

Algebraic expressions consist of variables, constants, and operations (such as addition, subtraction, multiplication, and division). They can represent real-world quantities, solve equations, or model problems. Understanding algebraic expressions is necessary for manipulating and solving them correctly.When working with algebraic expressions in the binomial expansion, considering terms like \(6a^{2.5}b^{0.5}\), there's a need to manage both numbers and variables efficiently:

• Variables represent unknowns and are crucial for forming expressions. In our example, terms are written in terms of \(a\) and \(b\).
• Understanding how to expand, condense, and simplify can uncover the essence of expressions and equations.
• During binomial expansion, the combination of coefficients and variables demands a careful handling of algebraic properties.

By mastering algebraic expressions, you prepare yourself for more advanced topics in mathematics, from solving quadratic equations to performing calculus operations. It lays the groundwork for logical thinking and problem-solving in varied mathematical fields.