Problem 68 - College Algebra and Calculus: An Applied Approach

Question

Simplify the expression.\(3 \sqrt{x+1}+10 \sqrt{x+1}\)

Step-by-Step Solution

Answer

The simplified expression is \( 13 \sqrt{x+1} \)

Step 1: Identify Like Terms

In the expression \(3 \sqrt{x+1}+10 \sqrt{x+1}\), the terms \(3 \sqrt{x+1}\) and \(10 \sqrt{x+1}\) are alike because they both contain the identical radical expression \( \sqrt{x+1} \).

Step 2: Combine Like Terms

When you have like terms that contain the same radical, you can add together their coefficients. Here, the coefficients of the like terms are 3 and 10. Their sum is \(3 + 10 = 13\).

Step 3: Write Final Answer

After adding the coefficients together, the simplified expression is \(13 \sqrt{x+1}\).

Key Concepts

Like Terms

Like terms are critical when simplifying expressions, and mastering their identification is a huge first step. Like terms are essentially terms that have the exact same variable parts, including any exponents or radicals. In mathematical expressions, these identical elements are what allow for the combination of terms.

An expression might read something like this: \(3x + 5x\). Both \(3x\) and \(5x\) are like terms because they each contain the variable \(x\) without any radicals or exponents distinguishing them.

Similarly, in a more complex expression, terms such as \(3 \sqrt{x+1}\) and \(10 \sqrt{x+1}\) are also like terms. They both feature the radical \(\sqrt{x+1}\), which is the exact same element in both terms. Recognizing like terms means identifying these shared pieces. This understanding is crucial for simplifying any type of algebraic expression efficiently.

Radical Expression

Radical expressions might seem intimidating at first, but they follow specific rules that can make your math journey smoother. A radical expression involves a root, most commonly a square root. When you see something like \(\sqrt{x+1}\), you are looking at a radical expression.

Understanding radicals involves grasping a few key concepts:

• The radical symbol \(\sqrt{}\) indicates the principal square root of the quantity inside.
• Things inside the radical are referred to as the "radicand". In \(\sqrt{x+1}\), "x+1" is the radicand.

When you simplify or manipulate radicals, ensure that any like radicals are combined just like you would with like terms.

For example, in the expression \(3 \sqrt{x+1}+10 \sqrt{x+1}\), the structure allows you to treat it just like combining regular numbers because the radicals are the same. Once you’re comfortable recognizing radicals and managing them according to rules of arithmetic, combining and simplifying expressions becomes much simpler.

Combining Coefficients

Combining coefficients is the core action when simplifying expressions that involve like terms. Coefficients are the numerical parts of expressions that are found directly before the variable or radical part.

Take, for instance, the terms \(3 \sqrt{x+1}\) and \(10 \sqrt{x+1}\). Here, 3 and 10 are the coefficients. To simplify, just add these coefficients while keeping the radical part unchanged.

• Add the coefficients directly, like \(3 + 10\), which results in 13.
• Attach this sum to the radical expression that both coefficients share, maintaining the integrity of the original terms.

This operation simplifies the expression to \(13 \sqrt{x+1}\). By always tackling the coefficients first and retaining the like term associated with them, you efficiently break down expressions, making them easier to manage and understand. The method becomes second nature with practice, easing your work across various mathematical problems.