Problem 51 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

Expansion of \( \left(x^{\frac{1}{3}}-x^{-\frac{1}{3}}\right)^{3}\) using binomial theorem results in a sum of four terms, each simplified to its smallest form.

Step 1: Recognize the Structure

Identify the terms of the binomial expression and the exponent by which it is raised. Here we have, \(a = x^{1/3}\), \(b = -x^{-1/3}\) and \(n = 3\)

Step 2: Apply the Binomial Theorem

The binomial theorem states that \((a + b)^n = \sum_{k = 0}^{n} {n \choose k} a^{n - k} b^{k}\). Applying that theorem, the expression becomes: \[ \sum_{k = 0}^{3} {3 \choose k} (x^{1/3})^{3-k} (-x^{-1/3})^{k}\]

Step 3: Compute the Coefficients

Calculate the binomial coefficients using the formula: \[ {n \choose k} = \frac{n!}{k!(n-k)!}\], where ! represents factorial. Using this formula, the expression now has 4 terms corresponding to \(k = 1, 2, 3, 4\).

Step 4: Simplify the Expression

With each term, simplify the powers and collect like-terms wherever possible. Final answer will be simplified form of sum of all four terms.

Key Concepts

Binomial Expansion

The concept of binomial expansion is closely tied to the Binomial Theorem, which provides a way to expand expressions raised to a power. It's incredibly useful when dealing with expressions of the form \((a + b)^n\). This theorem helps us break down complex expressions into simpler terms that are easier to handle.

The Binomial Theorem states that \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). Each term in the expansion is characterized by a coefficient \(\binom{n}{k}\), which represents the number of ways to choose \(k\) elements from \(n\).

• "\(a\)" is the first term in the binomial, which is raised gradually to decreasing powers.
• "\(b\)" is the second term, gradually raised to increasing powers.
• By summing over all values of \(k\), from 0 to \(n\), the full expansion is obtained.

In our original problem, the expansion is not simple due to the presence of fractional exponents in \(a = x^{1/3}\) and \(b = -x^{-1/3}\). Nevertheless, by keeping track of each term separately, the Binomial Theorem helps us derive the expanded expression, \(\sum_{k=0}^{3} {3 \choose k} (x^{1/3})^{3-k} (-x^{-1/3})^{k}\). This approach simplifies the job of expressing more complex algebraic expressions.

Simplifying Expressions

Simplifying expressions is a vital skill in algebra. It involves reducing expressions to their simplest form to make them easier to work with.

Once the binomial expansion provides us with several terms, the next step is breaking each of these terms into a simpler form. In our exercise, we expand the expression \(\left(x^{1/3}-x^{-1/3}\right)^{3}\) and produce terms that require additional simplification:

• Each term within the expansion can have powers of the variable \(x\) that must be simplified.
• Coefficients due to binomial expansion \({n \choose k}\) are calculated and applied.
• Negative exponents or roots can be replaced with their equivalent reciprocals or simplified completely when possible.

In the example, once you've calculated the individual terms for \(k = 0, 1, 2, 3\), simplifying means combining like terms and potentially using rules of exponents to remove divisors or negative powers of \(x\). The goal is to express the polynomial in its simplest possible form, both to reveal any symmetries or patterns and to make further algebraic manipulation easier.

Exponents in Algebra

Exponents are key components in algebra that represent repeated multiplication of a base number. They are a crucial part of simplifying and expanding algebraic expressions. Understanding exponents, including how to manipulate them, is essential when performing algebraic operations.

In mathematics, the rules for exponents are:

• \(a^m \cdot a^n = a^{m+n}\)
• \(\frac{a^m}{a^n} = a^{m-n}\)
• \( (a^m)^n = a^{m \cdot n}\)

These rules are fundamental, helping to simplify expressions by combining terms with the same base or breaking down complex powers. With fractional exponents, as seen in our exercise, these rules continue to apply:

• \(x^{1/3}\) indicates the cube root of \(x\).
• Negative exponents, like \(x^{-1/3}\), signify inverses or reciprocals.

Dealing with fractional and negative exponents requires special attention as they can complicate expressions significantly. By applying exponent rules consistently, such expressions can be made manageable and allow for a cleaner form, enabling clearer communication and calculation in algebraic problems.