Perform the indicated operations. See Examples 2–6. (7m+2n)^2

Verified step by step guidance

1. Recognize that the expression $(7m+2n{)}^2$ is a binomial squared, which can be expanded using the formula $(a+b{)}^2=a^2+2ab+b^2$.
2. Identify $a=7m$ and $b=2n$ in the expression $(7m+2n{)}^2$.
3. Calculate $a^2$ by squaring $7m$, which gives $(7m{)}^2=49m^2$.
4. Calculate $2ab$ by multiplying 2, $a=7m$, and $b=2n$, which gives $2\times 7m\times 2n=28mn$.
5. Calculate $b^2$ by squaring $2n$, which gives $(2n{)}^2=4n^2$.

Key Concepts

Binomial Expansion

Binomial expansion refers to the process of expanding expressions that are raised to a power, specifically in the form of (a + b)^n. The expansion can be achieved using the Binomial Theorem, which states that (a + b)^n = Σ (n choose k) * a^(n-k) * b^k, where k ranges from 0 to n. This theorem allows for systematic calculation of each term in the expansion.

Squaring a Binomial

Squaring a binomial involves applying the formula (a + b)^2 = a^2 + 2ab + b^2. In the context of the given expression (7m + 2n)^2, this means calculating the square of each term, adding twice the product of the two terms, and combining the results. This is a specific case of binomial expansion where n equals 2.

Coefficients in Algebra

Coefficients are numerical factors that multiply variables in algebraic expressions. In the expression (7m + 2n)^2, the coefficients are 7 and 2, which represent the multiplicative factors of the variables m and n, respectively. Understanding coefficients is essential for correctly applying operations like expansion and simplification in algebra.