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Chapter 15: Problem 3

Determine whether the expression can be simplified. $$5 \sqrt{3}+6 \sqrt{3}$$

Short Answer

Expert verified
The simplified form of the expression is \(11\sqrt{3}\)

Step by step solution

01

Identify like terms

In the given expression \(5 \sqrt{3}+6 \sqrt{3}\), both terms are the 'like terms' as they both have \(\sqrt{3}\) which acts as a common variable element
02

Add like terms

When you add like terms, you just add the coefficient (the number in front of the \(\sqrt{3}\)) of the terms. So, \(5 \sqrt{3} + 6 \sqrt{3} = (5+6) \sqrt{3} = 11\sqrt{3}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Like Terms
When simplifying radical expressions, identifying "like terms" is crucial. Like terms are components in an expression that feature the same radical part. In the example expression, \(5 \sqrt{3} + 6 \sqrt{3}\), both terms are considered like terms because the radical parts鈥擻(\sqrt{3}\)鈥攁re identical.

Unlike terms would have different radicals. Remember:
  • Only like terms can be combined in a simplification.
  • Look for the same numbers under the radical sign, such as \(\sqrt{3}\) or \(\sqrt{5}\).
Identifying like terms is the first step in simplifying any radical expression.
Coefficient
The coefficient in a radical expression is the number paired with the radical symbol. In the expression \(5 \sqrt{3} + 6 \sqrt{3}\), the coefficients are 5 and 6.

Coefficients function as the multiplier of the radicals. Here's a simple way to understand it:
  • The number 5 in \(5 \sqrt{3}\) is telling us we have 5 groups of \(\sqrt{3}\).
  • Similarly, the number 6 in \(6 \sqrt{3}\) means there are 6 groups of \(\sqrt{3}\).
To simplify, just add these coefficients together when dealing with like terms.
Radicals
A radical expression includes a symbol known as the radical symbol \(\sqrt{}\). It signifies a root operation鈥攎ost commonly a square root.

Radicals may appear complicated, but they follow straightforward rules, just like any other mathematical operation. In the expression \(5 \sqrt{3} + 6 \sqrt{3}\), \(\sqrt{3}\) is known as the "radicand," or the number inside the radical sign.

Here are a few important points to understand about radicals:
  • They operate under similar rules to variables in algebra.
  • They can only be combined if they have the same radicand.
  • Like terms among radicals combine by adding or subtracting the coefficients.
Understanding how radicals work is key to simplifying radical expressions.
Addition of Radicals
Adding radicals is similar to adding numbers with common variables. It requires that the radicals involved are like terms. The expression \(5 \sqrt{3}+6 \sqrt{3}\) involves addition of radicals.

To add radicals efficiently:
  • Ensure the radicand (the number under the radical) is identical.
  • Add only the coeffcients, while the radical part remains unchanged.
In our example, \(5 \sqrt{3} + 6 \sqrt{3}\) adds by summing up 5 and 6 which gives us \(11 \sqrt{3}\).

Just remember, you can't combine radicals with different radicands through simple addition. This method only works when the radicals are like terms.

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