Perform the indicated operations. (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 for the square of a sum: \((a + b)^2 = a^2 + 2ab + b^2\).
2. Identify the terms: here, \(a = 7m\) and \(b = 2n\).
3. Calculate the square of the first term: \(a^2 = (7m)^2 = 49m^2\).
4. Calculate twice the product of the two terms: \(2ab = 2 \times 7m \times 2n = 28mn\).
5. Calculate the square of the second term: \(b^2 = (2n)^2 = 4n^2\). Then, combine all parts to write the expanded expression as \(49m^2 + 28mn + 4n^2\).

Key Concepts

Binomial Expansion

Binomial expansion involves expressing the power of a binomial, such as (a + b)^2, as a sum of terms. For the square of a binomial, the formula is (a + b)^2 = a^2 + 2ab + b^2, which helps simplify expressions like (7m + 2n)^2.

Algebraic Operations with Variables

Understanding how to manipulate variables and coefficients is essential. This includes applying exponent rules, multiplying coefficients, and combining like terms when expanding expressions involving variables such as m and n.

Distributive Property

The distributive property states that a(b + c) = ab + ac. It is used to multiply each term inside the parentheses by the term outside or by each other in binomial multiplication, which is fundamental in expanding expressions like (7m + 2n)^2.