Problem 297 - Elementary Algebra

Question

In the following exercises, simplify. $$ (1+8 \sqrt{2})(1-8 \sqrt{2}) $$

Step-by-Step Solution

Answer

The simplified expression is \[ -127. \]

Step 1 - Identify the Expression

The given expression is \[ (1 + 8\sqrt{2})(1 - 8\sqrt{2}). \] This can be simplified using the difference of squares formula: \[ (a + b)(a - b) = a^2 - b^2. \]

Step 2 - Assign Values to a and b

Here, we can assign: \[ a = 1 \]and \[ b = 8 \sqrt{2} \]

Step 3 - Apply the Difference of Squares Formula

Substitute the values of a and b into the formula: \[ (1 + 8 \sqrt{2})(1 - 8 \sqrt{2}) = 1^2 - (8 \sqrt{2})^2. \]

Step 4 - Calculate the Squares

Compute the squares of the terms: \[ 1^2 = 1 \]and \[ (8 \sqrt{2})^2 = (8^2) (\sqrt{2}^2) = 64 \cdot 2 = 128. \]

Step 5 - Subtract the Results

Subtract the results of the squaring: \[ 1 - 128 = -127. \]

Key Concepts

difference of squares

The difference of squares is a powerful algebraic tool that simplifies many expressions. It relies on the identity: \( (a + b)(a - b) = a^2 - b^2 \). This formula states that the product of a sum and a difference of two terms equals the difference of their squares.
In our exercise, we used this formula to simplify the expression \((1 + 8\sqrt{2})(1 - 8\sqrt{2})\). By recognizing this pattern, we can assign \(a = 1\) and \(b = 8\sqrt{2}\).
This transforms our original problem into a simpler calculation: \( 1^2 - (8\sqrt{2})^2 \). This simplification greatly reduces the computational effort.

simplification

Simplification is the process of reducing an algebraic expression to its simplest form. This helps in making calculations easier and results more understandable.
In this exercise, the difference of squares formula plays a crucial role in simplification. By rewriting \((1 + 8\sqrt{2})(1 - 8\sqrt{2})\) into the form \(1^2 - (8\sqrt{2})^2 \), we make the expression manageable.
The next step involves calculating the squares: \(1^2 = 1 \) and \((8\sqrt{2})^2 = 128 \). The final step is to perform the subtraction: \( 1 - 128 = -127 \).
This step-by-step reduction shows how an initially complex expression can be boiled down to a simple numerical result.

algebraic expressions

Algebraic expressions are combinations of variables, numbers, and operations. Simplifying these expressions is fundamental in algebra to make them easier to work with.
The expression \( (1 + 8\sqrt{2})(1 - 8\sqrt{2}) \) is an example of how algebraic principles can simplify seemingly complex products. By using formulas and properties like the difference of squares, we turn these expressions into simpler forms.
Mastering the simplification of algebraic expressions keeps our solutions clear and concise, helping to understand and solve problems efficiently. As demonstrated, recognizing patterns and applying known formulas are key strategies in algebraic manipulation.