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Chapter 8: Problem 40

Simplify. $$\sqrt{3}+\sqrt{12}$$

Short Answer

Expert verified
\(3 \sqrt{3}\)

Step by step solution

01

Simplify Each Square Root

First, simplify the square roots individually. Note that \(\textstyle \sqrt{3} \) is already simplified, so let's focus on \(\textstyle \sqrt{12} \). We can break down \(\textstyle \sqrt{12} \) as follows: \(\textstyle \sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3} \).
02

Combine Like Terms

Next, add the simplified expressions. We already have \(\textstyle \sqrt{3} \), and now \(\textstyle 2\sqrt{3} \). Combine these like terms: \(\textstyle \sqrt{3} + 2\sqrt{3} \).
03

Add the Coefficients

Since both terms are like terms (they both contain \(\textstyle \sqrt{3} \)), add their coefficients: \(\textstyle 1 \sqrt{3} + 2\sqrt{3} = 3 \sqrt{3} \).

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Key Concepts

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

Combining Like Terms
In algebra, similar terms can often be combined to simplify expressions. When you have terms that share the same variable or root, they are known as 'like terms'.
This means they can be easily added or subtracted.
For instance, \( \textstyle \sqrt{3} + 2 \times \sqrt{3} \) can be combined because both terms have the \(\sqrt{3} \).
To combine these, simply add their coefficients.
Here, the coefficients are 1 and 2, so: \[1 \sqrt{3} + 2 \sqrt{3} = 3 \sqrt{3}\]
This technique can make solving algebraic expressions quicker and easier by reducing the number of terms.
Square Root Simplification
Square root simplification involves breaking down numbers inside square roots to their simplest form. To do this, you can factorize the number to identify perfect squares.
Take the example, \( \sqrt{12} \).
You can break it down as follows: \[ \sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \times \sqrt{3} = 2 \sqrt{3}\]
This process is essential because it allows you to combine like terms more easily.
By simplifying square roots, you can often see patterns and relationships that aren't obvious in their original forms.
Whenever you face a square root, check if it can be broken down further by identifying perfect squares.
Basic Algebra Operations
Basic algebra operations are foundational techniques that make solving equations possible. They include addition, subtraction, multiplication, and division.
For example, when you encounter an expression like \( \sqrt{3} + \sqrt{12} \),
follow these steps:
  • First, simplify any possible square roots to their simplest form, as shown with \(\sqrt{12}\).
  • Next, identify and combine like terms, such as adding coefficients for terms sharing \(\sqrt{3}\).

Finally, compute the operations: here, \[1 \sqrt{3} + 2 \sqrt{3} = 3 \sqrt{3}\].
Knowing these basic operations and applying them can make complex problems easier to solve.
Consistent practice with basic operations is key for mastering algebra and moving on to more advanced math topics.

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