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Magazine Chapter 13: Problem 33
Carry out the indicated expansions. $$(\sqrt{2}+\sqrt{3})^{5}$$
Short Answer
Expert verified
The expanded expression is \(109\sqrt{2} + 69\sqrt{3}\).
Step by step solution
01
Identify the Binomial Theorem
The expression \((\sqrt{2} + \sqrt{3})^5\) is a binomial expression raised to a power. We can use the Binomial Theorem, which states: \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). In this formula, \(a = \sqrt{2}\), \(b = \sqrt{3}\), and \(n = 5\).
02
Calculate Binomial Coefficients
To expand \((\sqrt{2} + \sqrt{3})^5\), we need to calculate the binomial coefficients \(\binom{5}{k}\) for \(k = 0, 1, 2, 3, 4, 5\). The coefficients are calculated as follows: - \(\binom{5}{0} = 1\)- \(\binom{5}{1} = 5\)- \(\binom{5}{2} = 10\)- \(\binom{5}{3} = 10\)- \(\binom{5}{4} = 5\)- \(\binom{5}{5} = 1\)
03
Expand Using the Binomial Theorem
Using the binomial coefficients, expand the expression: \((\sqrt{2} + \sqrt{3})^5 = \sum_{k=0}^{5} \binom{5}{k} (\sqrt{2})^{5-k} (\sqrt{3})^k\). Compute each term individually: 1. For \(k=0\), the term is \(1 \cdot (\sqrt{2})^5 \cdot (\sqrt{3})^0 = (\sqrt{2})^5\)2. For \(k=1\), the term is \(5 \cdot (\sqrt{2})^4 \cdot (\sqrt{3})^1\)3. For \(k=2\), the term is \(10 \cdot (\sqrt{2})^3 \cdot (\sqrt{3})^2\)4. For \(k=3\), the term is \(10 \cdot (\sqrt{2})^2 \cdot (\sqrt{3})^3\)5. For \(k=4\), the term is \(5 \cdot (\sqrt{2})^1 \cdot (\sqrt{3})^4\)6. For \(k=5\), the term is \((\sqrt{3})^5\)
04
Calculate Individual Powers and Terms
Now we calculate each of these terms: 1. \((\sqrt{2})^5 = \sqrt{32}\)2. \(5 \cdot (\sqrt{2})^4 \cdot (\sqrt{3}) = 5 \cdot 4 \sqrt{6} = 20\sqrt{6}\)3. \(10 \cdot (\sqrt{2})^3 \cdot (\sqrt{3})^2 = 10 \cdot 2\sqrt{18} = 20\sqrt{18}\)4. \(10 \cdot (\sqrt{2})^2 \cdot (\sqrt{3})^3 = 10 \cdot 2 \sqrt{27} = 20\sqrt{27}\)5. \(5 \cdot (\sqrt{2}) \cdot (\sqrt{3})^4 = 5 \cdot \sqrt{81} = 45\sqrt{2}\)6. \((\sqrt{3})^5 = 9\sqrt{3}\) After calculating the terms, simplify their radical components.
05
Simplify and Combine Like Terms
Simplify and add up all the terms:1. \(\sqrt{32} = 4\sqrt{2}\)2. \(20\sqrt{6}\) stays the same.3. \(20\sqrt{18} = 20 \cdot 3\sqrt{2} = 60\sqrt{2}\)4. \(20\sqrt{27} = 20 \cdot 3\sqrt{3} = 60\sqrt{3}\)5. \(45\sqrt{2}\) stays the same.6. \(9\sqrt{3}\) stays the same.Combine like terms:- \(4\sqrt{2} + 60\sqrt{2} + 45\sqrt{2} = 109\sqrt{2}\)- \(60\sqrt{3} + 9\sqrt{3} = 69\sqrt{3}\)
06
Write the Final Expression
The expanded expression is: \(109\sqrt{2} + 69\sqrt{3}\). This is the expansion of \((\sqrt{2} + \sqrt{3})^5\) using the binomial theorem.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Binomial Coefficients
Binomial coefficients play a crucial role in the expansion of binomial expressions like \((\sqrt{2} + \sqrt{3})^5\). They are represented as \(\binom{n}{k}\) and denote the coefficients for each term in the polynomial expansion. The binomial coefficient \(\binom{n}{k}\) is calculated using combinations, which essentially gives the number of ways to choose \(k\) elements from \(n\) elements regardless of the order.
Here鈥檚 how you calculate a binomial coefficient:
Here鈥檚 how you calculate a binomial coefficient:
- Use the formula: \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \)
- For example, to expand \((\sqrt{2} + \sqrt{3})^5\), compute: \(\binom{5}{0} = 1\), \(\binom{5}{1} = 5\), all the way to \(\binom{5}{5} = 1\).
Radical Expressions
Radical expressions involve roots, such as square roots, cube roots, etc. In our problem, both \(\sqrt{2}\) and \(\sqrt{3}\) are examples of radical expressions. Handling these properly during arithmetic operations is key to finding the correct expansion.
When working with radical expressions, remember:
When working with radical expressions, remember:
- \((\sqrt{x})^2 = x\): Squaring a square root returns the original number.
- Combine like terms by ensuring the terms have the same radical part. For example, \(3\sqrt{2} + 4\sqrt{2} = 7\sqrt{2}\).
Polynomial Expansion
Polynomial expansion using the Binomial Theorem aims to express the power of a binomial as a sum of terms. Each term consists of a product of a binomial coefficient, the first binomial term raised to a power, and the second binomial term also raised to a power.
Let鈥檚 walk through expanding \((\sqrt{2} + \sqrt{3})^5\) step by step:
Let鈥檚 walk through expanding \((\sqrt{2} + \sqrt{3})^5\) step by step:
- The general term is given by: \(\binom{n}{k} (a^{n-k} b^k)\), where \(a = \sqrt{2}\) and \(b = \sqrt{3}\).
- Each term in the expansion will involve calculating powers of the radical expressions \((\sqrt{2})^{5-k}\) and \((\sqrt{3})^k\), multiplied by the corresponding binomial coefficient.
