Question

Find the product. $$(11-6 x)^{2}$$

Short answer

The product for the expression \((11-6x)^{2}\) is \(121 - 132x + 36x^2\).

Step-by-step solution

Identify the Structure

The expression \((11-6x)^{2}\) is a perfect square in the form \((a-b)^{2}\). Here, \(a = 11\) and \(b = 6x\).

Apply the Formula

We will now apply the formula \((a-b)^2 = a^2 - 2ab + b^2\). Here, substitute \(a\) with 11 and \(b\) with 6x.

Calculate the Result

After substituting, we have \(11^2 - 2*11*6x + (6x)^2\). Simplifying this further, we have: \(121 - 132x + 36x^2\).

Key concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that includes numbers, variables, and operators. These expressions are the backbone of algebra and are used to describe relationships and changes. In our exercise, \( (11-6x)^{2} \) is an algebraic expression that encodes the area of a square with side \( 11-6x \). Such expressions require simplification to see the relationships between the components more clearly.

To work with algebraic expressions effectively, understanding the order of operations and the properties of exponents is essential. This not only helps in simplifying complex expressions but also lays the foundation for solving algebraic equations and inequalities. Simplification of algebraic expressions is a prerequisite for more advanced topics in algebra, including factoring, solving quadratic equations, and graphing linear equations.

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra that involves reducing expressions to their simplest form, making them easier to understand and work with. Simplification may include combining like terms, using distribution, and applying exponent rules.

In the context of our exercise, simplification entails applying the perfect square trinomial formula to expand and simplify the squared term \( (11-6x)^{2} \) into a polynomial. A key step is to remember that simplifying is not about guessing; it has precise rules. For instance, \( a^2 \) denotes the number or variable \( a \) multiplied by itself, whereas \( 2ab \) represents the product of twice \( a \) and \( b \). By following these rules, mistakes can be minimized and understanding improved.

Exponentiation

Exponentiation is an arithmetic operation, represented by an exponent, which tells us how many times a number, known as the base, is multiplied by itself. For example, \( a^2 \) means \( a \) multiplied by \( a \) once. It's a shorthand that makes representation of very large numbers or very small numbers manageable in algebra.

Understanding exponentiation is crucial when dealing with perfect square trinomials, as seen in our exercise. The square of a binomial, or any expression, is a case of exponentiation where the exponent is 2, indicating the expression is multiplied by itself once. So, \( (11-6x)^{2} \) becomes \( (11-6x)*(11-6x) \) before it is simplified further. Understanding this concept is crucial, as it can be extended to expressions with higher exponents, such as cubes or higher powers, which are common in algebraic functions and calculus.