Problem 108 - Elementary and Intermediate Algebra

Question

Look Alikes \(\cdots\) a. \(2 \sqrt{5}-\sqrt{5}+4 \sqrt{5}-6 \sqrt{5}\) b. \(2 \sqrt[3]{5}-\sqrt[3]{5}+4 \sqrt[3]{5}-6 \sqrt[3]{5}\)

Step-by-Step Solution

Answer

Both expressions simplify to \( -\sqrt{5} \) and \( -\sqrt[3]{5} \), respectively.

Step 1: Understand the Expression in Part (a)

In the expression \(2 \sqrt{5} - \sqrt{5} + 4 \sqrt{5} - 6 \sqrt{5}\), the terms \(2 \sqrt{5}\), \(-\sqrt{5}\), \(4 \sqrt{5}\), and \(-6 \sqrt{5}\) are multiples of \(\sqrt{5}\). This means we can factor out \(\sqrt{5}\) from these terms and simplify.

Step 2: Combine Like Terms in Part (a)

Combine the coefficients of \(\sqrt{5}\): - \(2 - 1 + 4 - 6\).Calculate the sum: \(2 - 1 = 1\), \(1 + 4 = 5\), \(5 - 6 = -1\).This results in \(-1 \sqrt{5}\).

Step 3: Simplify the Result in Part (a)

The simplified expression for Part (a) is \(-\sqrt{5}\).

Step 4: Understand the Expression in Part (b)

The expression \(2 \sqrt[3]{5} - \sqrt[3]{5} + 4 \sqrt[3]{5} - 6 \sqrt[3]{5}\) involves terms with the same base \(\sqrt[3]{5}\). We can factor out \(\sqrt[3]{5}\) from the terms and simplify.

Step 5: Combine Like Terms in Part (b)

Combine the coefficients of \(\sqrt[3]{5}\): - \(2 - 1 + 4 - 6\).Calculate the sum: \(2 - 1 = 1\), \(1 + 4 = 5\), \(5 - 6 = -1\).This results in \(-1 \sqrt[3]{5}\).

Step 6: Simplify the Result in Part (b)

The simplified expression for Part (b) is \(-\sqrt[3]{5}\).

Key Concepts

Combining Like Terms

When you encounter terms like those in the expressions given in the exercise, such as \(2 \sqrt{5}\), \(-\sqrt{5}\), \(4 \sqrt{5}\), and \(-6 \sqrt{5}\), you are dealing with like terms. *Like terms* are those that have the same variable and the same exponent. In this case, the variable is the entire radical, \(\sqrt{5}\), which acts as a consistent term among the given numbers.To simplify the expression, you should focus on the coefficients, which are the numerical factors that multiply the like term. You just need to add or subtract these coefficients. Let's go through the process step-by-step:

• Identify the like terms. Here, \(\sqrt{5}\) is the common term.
• Consider their coefficients: \(2, -1, 4,\) and \(-6\).
• Combine the coefficients: \(2 - 1 = 1\), \(1 + 4 = 5\), and finally \(5 - 6 = -1\).

The result, \(-1 \sqrt{5}\), is simplified because all like terms have been combined.
Combining like terms is a powerful tool in algebra that streamlines expressions and makes them smaller and easier to work with.

Radicals

Radicals are expressions that involve roots, like square roots or cube roots. In mathematical terms, a radical symbol \(\sqrt{}\) or \(\sqrt[3]{}\) is used to denote these roots.### Understanding Radicals

• The basic form is \(\sqrt[n]{x}\), where \(n\) is the index (or degree) of the root, and \(x\) is the radicand, which is the value under the radical sign.
• If no index is provided, it is understood to be a square root, meaning \(n = 2\).

In the exercise, \(\sqrt{5}\) stands for the square root of 5, while \(\sqrt[3]{5}\) refers to the cube root of 5.

### Simplifying RadicalsIn the context of simplification, radicals retain their form much like any variable, meaning you cannot usually simplify the radical itself unless it’s a perfect nth power. However, you can often simplify expressions involving radicals by combining like terms, as was done in this exercise.

Factoring Out Common Factors

Factoring is the process of breaking down expressions into multipliers that give the original expression when multiplied together. It's a useful technique when simplifying expressions.### Factoring ProcessIn algebra, when terms share a common factor, you can 'factor it out'. Here's how it applies to the given exercise:

• Notice that each term in the expressions \(2 \sqrt{5} - \sqrt{5} + 4 \sqrt{5} - 6 \sqrt{5}\) has a common factor of \(\sqrt{5}\).
• You can rewrite the expression by factoring \(\sqrt{5}\) out: \((2 - 1 + 4 - 6) \cdot \sqrt{5}\).
• Combine the coefficients within the parentheses, ultimately resulting in \(-1\), so you get \(-1 \cdot \sqrt{5}\) or simply \(-\sqrt{5}\).

This method not only simplifies the problem but also makes it more compact. Factoring is particularly useful in making equations easier to solve and plays a vital role in algebra where simplification is key.