Use the Binomial Theorem to expand each expression and write the result in simplified form. (x^1/3 +x^-1/3)³

Verified step by step guidance

1. Identify the binomial expression to be expanded: \(\left(x^{\frac{1}{3}} + x^{-\frac{1}{3}}\right)^3\).
2. Recall the Binomial Theorem formula for expansion: \(\left(a + b\right)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k\), where \(\binom{n}{k}\) is the binomial coefficient.
3. Set \(a = x^{\frac{1}{3}}\), \(b = x^{-\frac{1}{3}}\), and \(n = 3\). Write out each term of the sum for \(k = 0, 1, 2, 3\):
4. \[\binom{3}{0} a^3 b^0 + \binom{3}{1} a^2 b^1 + \binom{3}{2} a^1 b^2 + \binom{3}{3} a^0 b^3\]
5. Simplify each term by calculating the binomial coefficients and combining the powers of \(x\) using the rule \(x^m \cdot x^n = x^{m+n}\).

Key Concepts

Binomial Theorem

The Binomial Theorem provides a formula to expand expressions of the form (a + b)^n, where n is a non-negative integer. It uses binomial coefficients, often found in Pascal's Triangle, to determine the coefficients of each term in the expansion. This theorem simplifies the process of expanding powers of binomials without direct multiplication.

Exponents and Fractional Powers

Exponents indicate repeated multiplication, and fractional exponents represent roots; for example, x^(1/3) is the cube root of x. Understanding how to manipulate and simplify expressions with fractional and negative exponents is essential for correctly expanding and simplifying terms in the binomial expansion.

Simplification of Algebraic Expressions

After expanding using the Binomial Theorem, combining like terms and simplifying powers of variables is necessary. This involves applying exponent rules, such as adding exponents when multiplying like bases, and reducing expressions to their simplest form for clarity and correctness.