Problem 77 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Simplify the expression. $$ \frac{1}{\sqrt{a}}+\sqrt{a} $$

Step-by-Step Solution

Answer

The simplified expression is: \(\frac{\sqrt{a} + a}{\frac{1}{\sqrt{a}}}\).

Step 1: Identify the common denominator

For the two terms, \(\frac{1}{\sqrt{a}}\) and \(\sqrt{a}\), the common denominator is \(a\), because \(\sqrt{a} * \sqrt{a} = a\). We will rewrite both terms with the common denominator.

Step 2: Rewrite the terms with the common denominator

Rewrite the first term as \(\frac{1}{\sqrt{a}} = \frac{\sqrt{a}}{a}\) and the second term as \(\sqrt{a} = \frac{a}{\sqrt{a}}\).

Step 3: Combine the terms with the common denominator

Now that both terms have the same denominator, we can combine the numerators: \(\frac{\sqrt{a}}{a} + \frac{a}{\sqrt{a}} = \frac{\sqrt{a} + a}{\sqrt{a}}\).

Step 4: Convert the denominator back to the original form

Finally, rewrite the denominator from \(\sqrt{a}\) to \(\frac{1}{\sqrt{a}}\), which gives us: \(\frac{\sqrt{a} + a}{\frac{1}{\sqrt{a}}}\). So, the simplified expression is: \(\frac{\sqrt{a} + a}{\frac{1}{\sqrt{a}}}\).

Key Concepts

Understanding Common Denominator

When working with fractions and expressions that involve fractions, finding a common denominator is crucial. It's like finding a common language between two fractions so they can be added or subtracted easily. In the given problem, we have two different forms:

• A fraction: \( \frac{1}{\sqrt{a}} \)
• A radical next to a variable: \( \sqrt{a} \)

To simplify these expressions together, they need a shared base, or common denominator, they can agree upon. Here, \( \sqrt{a} * \sqrt{a} = a \), so the common denominator is \( a \). This conversion helps in harmonizing different forms so they can be summed or simplified together. Finding the common denominator allows us to balance these expressions by rewriting them in a way that's easier to combine.

Expression Simplification: Steps to Clarity

Simplifying expressions is all about breaking down complexities into manageable parts. In our exercise, we have the expression \( \frac{\sqrt{a} + a}{\frac{1}{\sqrt{a}}} \). The goal is to make this expression as simple as possible:

• First, both terms \( \frac{1}{\sqrt{a}} \) and \( \sqrt{a} \) are rewritten using the common denominator, \( a \).
• This involves expressing them in terms of a single base, which helps in combining them.
• Combining these numerators over the common denominator \( a \) leads to a simplified form.

It's akin to organizing various puzzle pieces into a coherent picture where every piece smoothly fits together. By simplifying the expression, not only does it become clearer, but it's also much easier to work with or solve further if necessary.

Rational Expressions Unveiled

Rational expressions are similar to fractions, but they involve variables in the numerator, the denominator, or both. They are an essential part of algebra because they provide a way to handle equations involving division that include variables. In this exercise:

• We started with a mix of a radical and a fraction.
• This combination is typical in rational expressions, where you often have to consolidate or divide complex parts systematically.

Rationalizing denominators or numerators, like in this case, is a frequent task.
By transforming \( \sqrt{a} \) suitably and ensuring our expression shares a common denominator, we deal with the whole rational expression more effectively. The pivot for handling rational expressions revolves around making the components compatible, leading to a simpler and more manageable form.