Question

Combine radicals, if possible. \(2 \sqrt{3}+\frac{\sqrt{3}}{2}\)

Short answer

Question: Simplify the expression \(2 \sqrt{3}+\frac{\sqrt{3}}{2}\). Answer: \(\frac{5\sqrt{3}}{2}\)

Step-by-step solution

Write down the given expression

First, let's write down the given expression. \(2 \sqrt{3}+\frac{\sqrt{3}}{2}\)

Find a common denominator

In order for two terms with different denominators to be added together, they must first have the same denominator. This requires finding the least common multiple (LCM) of their current denominators. Since the denominators are 1 and 2 (recall that \(2\sqrt{3}\) can be written as \(\frac{2\sqrt{3}}{1}\)), their LCM is 2.

Convert terms to have the common denominator

Now that we have identified that we need a common denominator of 2, let's convert the terms to have this denominator by multiplying the numerator and the denominator of the first radical term by 2: \(\frac{2\sqrt{3}}{1} \times \frac{2}{2} + \frac{\sqrt{3}}{2}\) This simplifies to: \(\frac{4\sqrt{3}}{2} + \frac{\sqrt{3}}{2}\)

Combine the terms

Now that both terms have the same denominator, we can combine them by adding the numerators together and keeping the denominator the same: \(\frac{4\sqrt{3} + \sqrt{3}}{2}\)

Simplify the result

Finally, we can simplify the expression by combining the numerators: \(\frac{5\sqrt{3}}{2}\) So, the simplified expression is \(\frac{5\sqrt{3}}{2}\).

Key concepts

Combining Radicals

When working with radical expressions, such as square roots, combining radicals involves adding or subtracting them. But there's a catch! You can only combine radicals if they have the same radicand, which is the number or expression inside the square root.
In our example, we have two terms:

• \(2\sqrt{3}\)
• \(\frac{\sqrt{3}}{2}\)

Both have the same radicand, \(\sqrt{3}\), which makes them eligible for combining. Once the terms are expressed with a common denominator, as shown in the solution steps, they can be added together. This results in a single term with a coefficient representing the sum of the individual coefficients.

Common Denominator

When adding fractions or terms that have different denominators, like those in our example, it's crucial to have a common denominator. This ensures that all terms are on equal footing and can be combined.
The process of "finding a common denominator" involves determining the smallest multiple that is common to the denominators of all terms involved. For our terms \(2\sqrt{3}\) (considered as \(\frac{2\sqrt{3}}{1}\)) and \(\frac{\sqrt{3}}{2}\), the least common multiple of 1 and 2 is 2. Once we've identified this number, we adjust each term so that they share the common denominator.

• Multiply the numerators and denominators of each term as needed to achieve the common denominator.
• This preserves the value of each term while making them compatible for addition or subtraction.

Simplifying Expressions

Simplifying expressions is the process of making them as clear and concise as possible. It's like cleaning up a room - you want everything neatly arranged and easy to understand.
After combining the radicals with a common denominator, we simplify by:

• Combining like terms, meaning those with the same base or radicand.
• Ensuring coefficients and constants are reduced to their simplest form.

In our worked solution, we see the expression \(\frac{4\sqrt{3} + \sqrt{3}}{2}\) simplified further. By adding the coefficients of \(\sqrt{3}\), we get \(\frac{5\sqrt{3}}{2}\).
This is our final, simplified expression, cleanly combining all components into a single, easily understandable form.