In Exercises 55–68, multiply using one of the rules for the square of a binomial. (aⁿ + 4bⁿ)²

Verified step by step guidance

1. Identify the expression as a binomial squared: $(a^n+4b^n{)}^2$.
2. Recall the formula for the square of a binomial: $(x+y{)}^2=x^2+2xy+y^2$.
3. Apply the formula to the given expression: let $x=a^n$ and $y=4b^n$.
4. Calculate $x^2=(a^n{)}^2=a^{2n}$.
5. Calculate $2xy=2(a^n)(4b^n)=8a^n{b}^n$ and $y^2=(4b^n{)}^2=16b^{2n}$.

Key Concepts

Binomial Expansion

Binomial expansion refers to the process of expanding expressions that are raised to a power, particularly binomials. The formula for the square of a binomial, (a + b)², is a² + 2ab + b². This concept is essential for simplifying expressions involving two terms and helps in understanding how to apply the rules of exponents.

Square of a Binomial

The square of a binomial is a specific case of binomial expansion where a binomial expression is multiplied by itself. The general formula is (x + y)² = x² + 2xy + y². In the given expression (aⁿ + 4bⁿ)², recognizing this pattern allows for the correct application of the formula to simplify the expression effectively.

Exponents and Powers

Exponents represent repeated multiplication of a base number. In the expression aⁿ and bⁿ, n indicates the power to which the base is raised. Understanding how to manipulate exponents, including the rules for multiplying and adding powers, is crucial for correctly expanding and simplifying expressions involving variables raised to powers.