Use the Binomial Theorem to expand each binomial and express the result in simplified form. (2a + b)⁶

Verified step by step guidance

1. Identify the binomial expression to expand: \((2a + b)^6\).
2. Recall the Binomial Theorem formula: \(\displaystyle (x + y)^n = \sum_{k=0}^n \binom{n}{k} x^{n-k} y^k\), where \(\binom{n}{k}\) is the binomial coefficient.
3. In this problem, set \(x = 2a\), \(y = b\), and \(n = 6\). The expansion will be \(\sum_{k=0}^6 \binom{6}{k} (2a)^{6-k} b^k\).
4. Calculate each term by finding the binomial coefficient \(\binom{6}{k}\), raising \$2a\( to the power \)(6-k)\(, and \)b\( to the power \)k\(, then multiply these values together for each \)k$ from 0 to 6.
5. Simplify each term by applying the exponent to \$2a$ (remember to raise 2 to the power as well), and write the full expanded expression as the sum of all these simplified terms.

Key Concepts

Binomial Theorem

The Binomial Theorem provides a formula to expand expressions of the form (x + y)^n, where n is a non-negative integer. It states that (x + y)^n equals the sum of terms C(n, k) * x^(n-k) * y^k, where C(n, k) are binomial coefficients. This theorem simplifies the expansion process without multiplying the binomial repeatedly.

Binomial Coefficients

Binomial coefficients, denoted as C(n, k) or "n choose k," represent the number of ways to choose k elements from a set of n elements. They appear as coefficients in the binomial expansion and can be calculated using factorials or Pascal's Triangle. These coefficients determine the weight of each term in the expanded expression.

Exponent Rules in Expansion

When expanding (2a + b)^6, each term involves powers of 2a and b whose exponents add up to 6. The exponent rules state that (xy)^n = x^n * y^n and (x^m)^n = x^(m*n). Applying these rules helps simplify each term by correctly raising coefficients and variables to their respective powers.