Problem 35 - Algebra 1: Concepts and Skills

Question

Simplify the expression using the sum and difference pattern. $$ (1+\sqrt{13})(1-\sqrt{13}) $$

Step-by-Step Solution

Answer

\(-12\)

Step 1: Identifying \(a\) and \(b\)

In the given expression \((1+\sqrt{13})(1-\sqrt{13})\), \(a = 1\) and \(b = \sqrt{13}\).

Step 2: Applying the Difference of Squares Formula

Substitute \(a\) and \(b\) into the formula \((a+b)(a-b) = a^2 - b^2\). So the simplified form becomes \(1^2 - (\sqrt{13})^2\).

Step 3: Simplifying the Expression

Finally, simplify \(1^2 - (\sqrt{13})^2\) to \(1 - 13\), which equals \(-12\).

Key Concepts

Sum and Difference Pattern

The sum and difference pattern is a powerful tool in algebra. It involves expressions that are structured in the form

• \((a+b)(a-b)\)

Why is this pattern helpful? Because it simplifies our work by using the difference of squares.
Essentially, this pattern helps us find products quickly without having to multiply each term individually.
When using the sum and difference pattern:

• Identify which terms represent \(a\) and \(b\).
• Apply the formula: \((a+b)(a-b) = a^2 - b^2\).

This approach saves time and makes the expression easier to handle. In our example problem, by recognizing that \(1\) is \(a\) and \(\sqrt{13}\) is \(b\), we can directly apply the formula to reach the solution.

Algebraic Expressions

Understanding algebraic expressions is crucial to mastering algebra. Algebraic expressions are combinations of variables, numbers, and operations. They can be as simple as

• \(a + b\)

Or more complex like

• \((1+\sqrt{13})(1-\sqrt{13})\)

In these expressions, each component plays a role:

• Variables represent unknown values.
• Constants are fixed numbers.
• Operations dictate how the numbers and variables interact.

Understanding these components helps in breaking down problems and making sense of them. Working smoothly with algebraic expressions requires familiarity with these parts and the ability to manipulate them using rules like the sum and difference pattern.

Simplifying Expressions

Simplifying expressions is key to solving algebra problems efficiently. When we simplify, we aim to reduce expressions to their simplest form. This makes them easier to work with and understand.To simplify expressions, follow these steps:

• Apply algebraic rules, such as the sum and difference pattern.
• Perform any arithmetic calculations, such as squaring numbers or roots.
• Combine like terms if applicable.

In the provided exercise, by calculating \(1^2 - (\sqrt{13})^2\), we are simplifying \(a^2 - b^2\).
This breaks down to \(1 - 13\), leading to the simplified answer of \(-12\).
Simplifying expressions not only gives us the final answer but also builds a deeper understanding of mathematical principles.
Efficient simplification can also prevent errors and enhances problem-solving capabilities.