Problem 99 - Algebra 2

Question

Simplify. Rationalize all denominators. $$ (\sqrt{5}-1)(\sqrt{5}+4) $$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(1 + 3\sqrt{5}\).

Step 1: Distribute

Distribute the \(\sqrt{5}\) and -1 across the \(\sqrt{5}\) and +4, hence multiplying the terms within the parentheses. This gives: \[\sqrt{5} * \sqrt{5} + \sqrt{5} * 4 - 1 * \sqrt{5} - 1 * 4\]

Step 2: Simplify

Substitute and simplify. The square root of a number squared is the number itself. That is, \(\sqrt{5} * \sqrt{5} = 5\). Therefore, the expression simplifies to: \[5 + 4\sqrt{5} - \sqrt{5} - 4\]

Step 3: Collect like terms

Next, combine the like terms, which are \(4\sqrt{5}\) and \(-\sqrt{5}\) to simplify the above expression to: \[1 + 3\sqrt{5}\]

Key Concepts

Simplification

Simplification is the process of reducing an expression to its simplest form.
In our exercise, we need to perform several steps to reach the simplest form of the given expression. Each step requires careful calculations to ensure that each part of the expression is fully simplified before moving to the next stage.In the expression \((\sqrt{5}-1)(\sqrt{5}+4)\), we start by distributing each term and then simplifying the resulting expression.
The goal of simplification is to make the expression as concise as possible, while ensuring that it is mathematically equivalent to the original. In this exercise, we simplified by:

• Squaring the square root: \(\sqrt{5} * \sqrt{5} = 5\)
• Combining terms that are not like terms into a more straightforward sum

Ultimately, simplification helps us understand the problem more easily and often brings us to an aesthetically neater solution.

Distribution

Distribution involves expanding an expression by multiplying each term in one part of the expression by each term in another. In mathematical terms, it's like using the distributive property, which dictates: For two binomials like \((a+b)(c+d),\) you distribute each term across the others as follows: \(a*c + a*d + b*c + b*d.\)In our exercise, distribution was applied to \((\sqrt{5}-1)(\sqrt{5}+4).\) Here's how the process worked:

• Multiply \(\sqrt{5}\) by both \(\sqrt{5}\) and \(+4\)
• Multiply \(-1\) by both \(\sqrt{5}\) and \(4\)

This results in an expanded expression: \(\sqrt{5}*\sqrt{5} + \sqrt{5}*4 - 1*\sqrt{5} - 1*4,\) providing a clear view of how each component interacts. The distribution step is crucial for decomposing polynomials and simplifying expressions.

Combining Like Terms

Combining like terms is an essential step in finalizing the simplification of any algebraic expression. "Like terms" are terms that contain the same variables raised to the same power. In our case, the like terms involve the square root of 5, \(\sqrt{5}.\)After distribution and initial simplification, our expression became \(5 + 4\sqrt{5} - \sqrt{5} - 4.\) To simplify further, we identified and combined the terms involving \(\sqrt{5},\) as follows:

• The terms \(4\sqrt{5}\) and \(-\sqrt{5}\) are like terms because they both have the factor \(\sqrt{5}.\)
• Subtracting \(-\sqrt{5}\) from \(4\sqrt{5}\) yields \(3\sqrt{5}.\)

Combining like terms in this way reduces clutter and simplifies our expression to \(1 + 3\sqrt{5}.\)Through this process, the expression is much more manageable and reflects the cleanest form possible, making it easier to understand and work with.