Writing (2a - b)^{3} in Expanded Form
When dealing with algebraic expressions raised to a power, it is often necessary to expand the expression to reveal the individual terms. This process is known as the binomial expansion, and it can be applied to the expression (2a - b)^{3}.
To expand the expression (2a - b)^{3}, we can use the binomial theorem, which states that for a binomial expression (a + b)^n, the expanded form can be written as:
(a + b)^n = ∑ (n choose k) * a^(n-k) * b^k, where k ranges from 0 to n.
In our case, the binomial expression is (2a - b), and the power is 3. Therefore, we can apply the binomial theorem as follows:
(2a - b)^3 = (2a)^3 - 3(2a)^2(b) + 3(2a)(b)^2 - (b)^3
Let's break down the calculation step by step:
- (2a)^3 = 8a^3
- -3(2a)^2(b) = -12a^2b
- 3(2a)(b)^2 = 12ab^2
- -(b)^3 = -b^3
Combining these terms, we get the final expanded form:
(2a - b)^3 = 8a^3 - 12a^2b + 12ab^2 - b^3
This expression represents the cube of the binomial (2a - b), with each term obtained by applying the binomial theorem.
By expanding the expression (2a - b)^{3} in this way, we can clearly see the individual terms and their coefficients, making it easier to work with and manipulate the expression in various mathematical contexts.