Use the Binomial Theorem to expand each expression and write the result in simplified form. (x³ +x^-2)⁴

Verified step by step guidance

1. Identify the binomial expression to be expanded: \(\left(x^{3} + x^{-2}\right)^4\).
2. Recall the Binomial Theorem formula: \(\left(a + b\right)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k}\), where \(\binom{n}{k}\) is the binomial coefficient.
3. Apply the formula with \(a = x^{3}\), \(b = x^{-2}\), and \(n = 4\). Write the sum as \(\sum_{k=0}^{4} \binom{4}{k} (x^{3})^{4-k} (x^{-2})^{k}\).
4. Simplify each term inside the sum by applying the power of a power rule: \((x^{3})^{4-k} = x^{3(4-k)}\) and \((x^{-2})^{k} = x^{-2k}\). Then combine the powers of \(x\) by adding exponents: \(x^{3(4-k) + (-2k)}\).
5. Write out each term explicitly for \(k=0\) to \(k=4\) using the binomial coefficients \(\binom{4}{k}\), simplify the powers of \(x\), and then sum all terms to get the expanded expression.

Key Concepts

Binomial Theorem

The Binomial Theorem provides a formula to expand expressions raised to a power, such as (a + b)^n. It states that (a + b)^n equals the sum of terms C(n, k) * a^(n-k) * b^k, where C(n, k) are binomial coefficients. This theorem simplifies the expansion process without multiplying the expression repeatedly.

Binomial Coefficients

Binomial coefficients, denoted as C(n, k) or "n choose k," represent the number of ways to choose k elements from n. They appear as coefficients in the binomial expansion and can be found using Pascal's Triangle or the formula C(n, k) = n! / (k!(n-k)!). These coefficients determine the weight of each term in the expansion.

Handling Negative and Fractional Exponents

When expanding expressions like (x^3 + x^-2)^4, it is important to correctly apply exponent rules. Negative exponents indicate reciprocals (x^-2 = 1/x^2), and when multiplying powers with the same base, exponents add. Simplifying each term after expansion requires careful management of these exponents to write the final expression in simplest form.