Problem 54 - Intermediate Algebra

Question

Simplify by combining like radicals. $$ 11 \sqrt{3}+2 \sqrt{3} $$

Step-by-Step Solution

Answer

The simplified expression is \(13 \sqrt{3}\).

Step 1: Identify the Like Radicals

In this expression, \(11 \sqrt{3}\) and \(2 \sqrt{3}\) are like radicals because they both share the same radical part \(\sqrt{3}\). This means they can be combined.

Step 2: Combine the Coefficients

Since the radicals are alike, you can add the coefficients (numbers in front of the square root). Add the coefficients: \(11 + 2 = 13\). Keep the \(\sqrt{3}\) part the same.

Step 3: Write the Simplified Expression

The simplified expression is \(13 \sqrt{3}\). We combined the coefficients and retained the like radical part from both terms.

Key Concepts

Understanding Radicals

Radicals are mathematical symbols used to represent roots, such as square roots or cube roots, of numbers. The symbol for a square root is \(\sqrt{}\). A radical expression is any expression that contains a radical symbol. In the context of our exercise, both \(11\sqrt{3}\) and \(2\sqrt{3}\) are radical expressions.
One important aspect of radicals is that the number inside the square root, known as the radicand, dictates whether two radicals are "like" radicals. Like radicals have the same radicand. In our exercise, both terms have \(\sqrt{3}\) as their radicand, making them like radicals. Thus, they can be simplified.
Radicals can often be intimidating, but remember they just serve the purpose of indicating roots. When simplifying them, especially in algebraic expressions, our main focus is on the coefficient—the number outside the radical symbol.

Combining Like Terms

Combining like terms is a technique used to simplify algebraic expressions. When radicals have the same radicand ("like radicals"), we treat the radicals much like algebraic terms and combine their coefficients.
This means if two terms in an expression have the same radical part, their coefficients can be added or subtracted just like any other like variables. For instance, in our example:

• Two terms are \(11\sqrt{3}\) and \(2\sqrt{3}\).
• Since the radicals \(\sqrt{3}\) are the same, we can simply add the coefficients \(11\) and \(2\).

The result is \(13\sqrt{3}\). The radical part remains unchanged while only the coefficients are combined.

Intermediate Algebra

Intermediate algebra involves a variety of mathematical operations and simplification processes used to solve more complex problems. Simplifying expressions with radicals is a common type of problem in intermediate algebra.
This topic often involves the rules of exponents, manipulating algebraic terms, and applying the distributive property when necessary. Being comfortable with simplifying like radicals is an essential skill.
In the case of our exercise, the process of identifying like radicals—those with identical radicands—and simplifying them by combining coefficients is crucial. It helps prepare students for more advanced algebraic expressions and equations where these operations need to be performed regularly.

• This entails a keen eye for identifying similarities in terms.
• Also, familiarizing yourself with combining numeric and algebraic coefficients.

Being proficient in these skills supports success in more challenging algebra courses and mathematical problems.