Problem 52 - Algebra and Trigonometry

Question

Use the Binomial Theorem to expand each expression and write the result in simplified form. $$ \left(x^{\frac{2}{3}}-\frac{1}{\sqrt[3]{x}}\right)^{3} $$

Step-by-Step Solution

Answer

The simplified form is \(x^2 - 3x^{1/3} + 3x^{-1/3} - 1\)

Step 1: Recall the Binomial Theorem

The Binomial Theorem states that, \((a + b)^n = \sum_{{k=0}}^{n} C(n, k) * a^{n-k} * b^{k}\), where \(C(n, k)\) are the Binomial coefficients. These are calculated as \(C(n, k) = n! / [(n-k)!k!]\), where '!' denotes factorial.

Step 2: Apply the Binomial Theorem to our expression

The expression can be seen as \((a - b)^3\) where \(a = x^{2/3}\) and \(b = 1/\sqrt[3]{x}\). Applying the Binomial Theorem gives us 4 terms: term1 = \((C(3, 0) \cdot (a^{3-0}) \cdot (-b)^0)\), term2 = \((C(3, 1) \cdot (a^{3-1}) \cdot (-b)^1)\), term3 = \((C(3, 2) \cdot (a^{3-2}) \cdot (-b)^2)\), term4 = \((C(3, 3) \cdot (a^{3-3}) \cdot (-b)^3)\)

Step 3: Calculate each term

Term1 = \(1\cdot x^2 = x^2\), Term2 = \(-3\cdot x^{2/3} \cdot 1/\sqrt[3]{x} = -3x^{2/3 - 1/3} = -3x^{1/3}\), Term3 = \(3\cdot x^{2/3} \cdot (1/x) = 3x^{2/3 - 3/3} = 3x^{-1/3}\), Term4 = \(-1\cdot 1 = -1\)

Step 4: Add up all terms

Finally, by adding up all the terms, we obtain the expanded expression as: \(x^2 - 3x^{1/3} + 3x^{-1/3} - 1\)

Key Concepts

Binomial Coefficients

Understanding binomial coefficients is essential when studying the Binomial Theorem. These coefficients, represented as \( C(n, k) \), appear in the expansion of a binomial expression, \((a + b)^n\), and are crucial for determining the coefficients that accompany each term.

To calculate a binomial coefficient, you use the formula \( C(n, k) = \frac{n!}{(n-k)!k!} \). The exclamation point '!' denotes a factorial, meaning you multiply all positive integers from 1 up to that number. For instance, \(4! = 1 \times 2 \times 3 \times 4 = 24\).

In our exercise, the binomial coefficients are found within the expression \((x^{2/3} - 1/\sqrt[3]{x})^3\), which determines how many times each term will be multiplied. Simplifying expressions like this involves calculating binomial coefficients for each term and ensuring the correct application of the theory.

For example, the coefficient of the second term in the expansion is \( C(3, 1) \), which can be calculated as \(\frac{3!}{(3-1)! \times 1!} = 3\). This number becomes the coefficient for the second term after applying the minus sign associated with the binomial's second element.

Factorials in Algebra

When dealing with binomial expansions, factorials in algebra frequently arise. A factorial, represented by the symbol '!', is imperative for calculating combinations and permutations, which lean heavily on factorial computations.

In algebra, we use factorials to find out how many ways things can be organized or chosen. For instance, the binomial coefficient formula relies on factorial calculations. In the context of our exercise, you encountered factorials while determining the binomial coefficients.

It's also important to consider the properties of factorials, especially zero factorial, which is defined as \(0! = 1\). This definition might seem odd but is consistent within the framework of combinatorics and allows for the smooth application of binomial theorem formulas.

Practical understanding of how factorials work supports the simplification of algebraic expressions—which often confuses students. By recognizing patterns within factorial calculations, students can simplify seemingly complex expressions within the binomial theorem expansion.

Algebraic Expressions Simplification

Algebraic expressions simplification is a fundamental skill in mathematics. It involves reducing expressions to their simplest form by performing operations like addition, subtraction, and exponentiation according to the order of operations.

In our exercise, simplifying the expanded expression \(x^2 - 3x^{1/3} + 3x^{-1/3} - 1\) is the final step of applying the binomial theorem. Simplification makes it easier to understand and manipulate the algebraic expression, and it allows us to possibly solve equations or inequalities more efficiently.

To simplify algebraic expressions, we combine like terms, which are terms with the same variable raised to the same power. In the exercise, notice how exponents are managed when multiplying terms, by adding or subtracting the exponents based on the laws of exponents. For instance, \(x^{2/3} \times x^{-1/3} = x^{2/3 - 1/3}\).

Mastering the simplification process requires familiarity with the properties of exponents, the distributive property, and the associative and commutative properties of addition and multiplication. Recognizing how these properties apply in the Binomial Theorem helps bridge the gap between theory and practice, moving from the step by step calculation of each term to the final, simplified algebraic expression.