Problem 61 - Elementary and Intermediate Algebra

Question

Perform the operations. $$ (4 f+0.4)(4 f-0.4) $$

Step-by-Step Solution

Answer

The result is \(16f^2 - 0.16\).

Step 1: Understand the Form of the Expression

The expression \((4f+0.4)(4f-0.4)\) is in the form \((a+b)(a-b)\), which is a difference of squares.

Step 2: Apply the Difference of Squares Formula

The difference of squares formula is \((a+b)(a-b) = a^2 - b^2\). Identify \(a = 4f\) and \(b = 0.4\). Then, apply the formula.

Step 3: Calculate \(a^2\)

Calculate \(a^2 = (4f)^2 = 16f^2\).

Step 4: Calculate \(b^2\)

Calculate \(b^2 = (0.4)^2 = 0.16\).

Step 5: Substitute and Simplify

Substitute \(a^2\) and \(b^2\) back into the formula to obtain the simplified expression: \( (4f)^2 - (0.4)^2 = 16f^2 - 0.16 \).

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and mathematical operations. In the expression \((4f+0.4)(4f-0.4)\), "4f" is a term where "4" is the coefficient and "f" is the variable. The expression also includes constants like "0.4".

An algebraic expression can have:

• Constants – numbers that do not change
• Variables – letters that can represent changing numbers
• Operators – symbols that show mathematical operations such as addition and subtraction

When you encounter expressions like \((4f+0.4)(4f-0.4)\), it is helpful to recognize patterns like the difference of squares, which can make it easier to simplify them.

Factoring: Use the Difference of Squares

Factoring is a method used to write an expression as a product of its factors. Recognizing the form of an expression can help you in factoring it efficiently. The given expression \((4f+0.4)(4f-0.4)\) fits the pattern of a difference of squares. This pattern is expressed as \((a+b)(a-b) = a^2 - b^2\).

Here's how it works:

• Identify the square roots of the first and last terms in the expression. In our case, \(a = 4f\) and \(b = 0.4\).
• Apply the formula by substituting \(a\) and \(b\), which will yield \(a^2 - b^2\).
• Calculate the squares: \((4f)^2 = 16f^2\) and \((0.4)^2 = 0.16\).

This simplifies to \(16f^2 - 0.16\), effectively factoring the expression.

Simplification of Expressions

Simplification involves reducing an expression to its simplest form. This is the process you're doing when you convert \((4f+0.4)(4f-0.4)\) into \(16f^2 - 0.16\).

Steps to simplify using the difference of squares:

• Apply the difference of squares formula: \((a+b)(a-b) = a^2 - b^2\).
• Substitute the calculated values of \(a^2\) and \(b^2\) into the equation.
• Simplify the resulting expression if necessary. In this case, \(16f^2 - 0.16\) is already simplified.

Always check if you can reduce the expression further by combining like terms or factoring common factors. This practice keeps expressions neat and as elegant as possible for easier interpretation and use.