Problem 90 - College Algebra with Modeling and Visualization

Question

Multiply and simplify. $$ (5-\sqrt{5})(5+\sqrt{5}) $$

Step-by-Step Solution

Answer

The simplified result is 20.

Step 1: Recognize the Difference of Squares Formula

The expression \((5 - \sqrt{5})(5 + \sqrt{5})\) is in the form of \((a-b)(a+b)\), which is a difference of squares. The difference of squares formula states that \((a-b)(a+b) = a^2 - b^2\).

Step 2: Identify the Values of 'a' and 'b'

In the expression \((5 - \sqrt{5})(5 + \sqrt{5})\), identify \(a = 5\) and \(b = \sqrt{5}\).

Step 3: Substitute into the Formula

Using the values \(a = 5\) and \(b = \sqrt{5}\) in the formula \(a^2 - b^2\), substitute these values to get \(5^2 - (\sqrt{5})^2\).

Step 4: Calculate Each Square

Calculate \(5^2\) which equals 25, and \((\sqrt{5})^2\) which equals 5.

Step 5: Subtract the Squares

Subtract the calculated squares: 25 (\(a^2\)) minus 5 (\(b^2\)) to get the final answer. So, 25 - 5 equals 20.

Key Concepts

Difference of Squares

The multiplication of \((5-\sqrt{5})(5+\sqrt{5})\) is a classic example of a mathematical concept called the **difference of squares**.This idea helps in simplifying expressions that look like the product of two binomials.Here’s how it works:

• When you see the format \((a-b)(a+b)\), you can recognize it as a difference of squares.
• The formula for the difference of squares is \(a^2-b^2\). It means, rather than having to multiply each term out individually, you can directly simplify by subtracting the square of one term from the other.

In this case, \(a = 5\) and \(b = \sqrt{5}\). When you apply the formula by squaring these terms and then subtracting, you find that:\[ 5^2 - (\sqrt{5})^2 = 25 - 5 = 20 \].This powerful method showcases the elegance and efficiency of algebra.

Square Roots

Square roots are essential in algebra and play a vital role in many mathematical calculations.A square root \(\sqrt{x}\) is a number that produces \(x\) when multiplied by itself.For example, the square root of \(25\) is \(5\), because \(5 \times 5 = 25\).
When dealing with expressions involving square roots, such as \((5-\sqrt{5})(5+\sqrt{5})\), it is crucial to understand how to work with square roots:

• When you square a number like \(\sqrt{5}\), you revert it back to \(5\). This is because \((\sqrt{5})^2 = 5\).
• Knowing this property of square roots helps simplify expressions and solve equations effectively.

In our exercise, recognizing and handling the square root correctly was the key to arriving at the simplified result of \(20\).

Simplification

Simplification in algebra is the process of making expressions as concise as possible.By reducing expressions to their simplest form, you can solve problems more efficiently and accurately.
In our example, we simplified \((5-\sqrt{5})(5+\sqrt{5})\) using algebraic concepts like the difference of squares.Here are the key steps in simplification:

• Identify mathematical patterns, such as the difference of squares, to facilitate simplification.
• Perform arithmetic operations precisely, like squaring terms and subtracting results.
• Re-confirm the calculations to ensure that the expression is at its simplest form. In this exercise, calculation confirmed that \(25 - 5 = 20\).

Simplification doesn’t just make expressions easier to handle. It also enhances our understanding and provides deeper insights into the methods and rules of algebra.