Find each product. [(3a+b)-1]²

Verified step by step guidance

1. Recognize that the expression is a square of a binomial: \([(3a+b)-1]^2\). This can be rewritten as \((3a + b - 1)^2\).
2. Recall the formula for the square of a trinomial: \((x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2xz + 2yz\).
3. Identify the terms: \(x = 3a\), \(y = b\), and \(z = -1\).
4. Apply the formula by squaring each term and adding twice the product of each pair: \( (3a)^2 + b^2 + (-1)^2 + 2(3a)(b) + 2(3a)(-1) + 2(b)(-1) \).
5. Simplify each term and combine like terms to write the expanded form of the product.

Key Concepts

Exponent Rules

Exponent rules govern how to handle powers in algebraic expressions. For example, a negative exponent indicates the reciprocal of the base raised to the positive exponent, so (x)^-1 = 1/x. Understanding these rules is essential for simplifying expressions like [(3a + b)^-1]^2.

Binomial Expressions

A binomial is an algebraic expression with two terms, such as (3a + b). Recognizing binomials helps in applying operations like squaring or multiplying, which often involve using formulas like (x + y)^2 = x^2 + 2xy + y^2.

Power of a Power Property

The power of a power property states that (x^m)^n = x^(m*n). This means when raising a power to another power, you multiply the exponents. Applying this property simplifies expressions like [(3a + b)^-1]^2 to (3a + b)^-2.