Problem 36 - Algebra A Combined Function

Question

Add or subtract as indicated. Assume that all variables represent positive real numbers. $$ \frac{\sqrt{45}}{10}+\frac{7 \sqrt{5}}{10} $$

Step-by-Step Solution

Answer

The solution is \( \sqrt{5} \).

Step 1: Simplifying the First Term

The expression is \( \frac{\sqrt{45}}{10} + \frac{7\sqrt{5}}{10} \). First, simplify \( \sqrt{45} \). Since \( 45 = 9 \times 5 \), we have \( \sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5} \). Thus, \( \frac{\sqrt{45}}{10} = \frac{3\sqrt{5}}{10} \).

Step 2: Combining Like Terms

Now, the expression becomes \( \frac{3\sqrt{5}}{10} + \frac{7\sqrt{5}}{10} \). Since both terms have \( \sqrt{5} \) in the numerator, we can add the coefficients. The coefficients are 3 and 7, so we add them to get: \( \frac{10\sqrt{5}}{10} \).

Step 3: Simplifying the Final Expression

We can simplify \( \frac{10\sqrt{5}}{10} \) by canceling out the common factor of 10 in the numerator and the denominator. This leaves us with \( \sqrt{5} \).

Key Concepts

Simplifying Radicals

The process of simplifying radicals is quite straightforward once you get the hang of it. It involves breaking down the expression inside the square root into its prime factors and then simplifying from there. For example, with \( \sqrt{45} \):

• First, recognize that 45 can be broken down into 9 and 5, because \( 9 \times 5 = 45 \).
• Since 9 is a perfect square (\( 3^2 \)), we can take the square root of 9 out of the radical.
• This simplifies \( \sqrt{45} \) to \( 3\sqrt{5} \), because \( \sqrt{9} \) equals 3.

Always check for the largest perfect square factor within the radical. This will help simplify the expression further and make it easier to manage in later calculations. It's a critical first step before moving on to more complex operations with radicals.

Combining Like Terms

Combining like terms, especially when dealing with radicals, follows a similar logic to combining like terms in polynomial expressions. To simplify, focus on the **coefficients** of the expressions that share the same radical part. Let's break it down in our example:

• Both terms in our expression \( \frac{3\sqrt{5}}{10} + \frac{7\sqrt{5}}{10} \) include \( \sqrt{5} \) in the numerator.
• Because they share the same radical, it's permissible—and indeed necessary—to combine their coefficients (3 and 7).
• Add the coefficients together: \( 3 + 7 = 10 \).
• This results in a single term: \( \frac{10\sqrt{5}}{10} \).

Combining like terms is a key operation that simplifies your calculation process and helps to bring clarity to complex expressions. The clearer and simpler each step is, the easier it is to solve correctly.

Adding and Subtracting Radicals

Adding and subtracting radicals is quite similar to adding and subtracting regular numbers, under one condition: the radicals (numbers under the root sign) must be the same.

In this operation, first make sure you have simplified all radical expressions, as outlined under "Simplifying Radicals." Once that’s done, follow these guidelines:

• If the radical parts (e.g., \( \sqrt{5} \)) match, you can add or subtract the coefficients directly.
• If the radicals differ, they cannot be directly combined.

In our original problem, after combining like terms, we ended up with \( \frac{10\sqrt{5}}{10} \). To complete the simplification:

• The coefficient 10 in the numerator and the 10 in the denominator cancel each other out.
• This gives us the final answer: \( \sqrt{5} \).

Grasping the rule that you can only combine radicals with the same indices is essential. Once you know this, adding and subtracting becomes a much faster and more intuitive process. Always start by simplifying and end by looking for opportunities to combine terms.