Problem 187 - Elementary Algebra

Question

In the following exercises, simplify. $$ 2 \sqrt{12}+3 \sqrt{48} $$

Step-by-Step Solution

Answer

The simplified form is \( 16 \sqrt{3} \).

Step 1 - Simplify the square roots

First, simplify the square roots inside each term. For \( \sqrt{12} \) we have: \(\sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2 \sqrt{3}\). For \(\sqrt{48} \) we have: \(\sqrt{48} = \sqrt{16 \cdot 3} = \sqrt{16} \cdot \sqrt{3} = 4 \sqrt{3}\).

Step 2 - Substitute simplified square roots into the expression

Replace each square root term with its simplified form in the expression \(2 \sqrt{12} + 3 \sqrt{48} \): \( 2 \cdot 2 \sqrt{3} + 3 \cdot 4 \sqrt{3} \)

Step 3 - Multiply coefficients

Now multiply the coefficients with the simplified square roots: \( 2 \cdot 2 \sqrt{3} = 4 \sqrt{3} \) \( 3 \cdot 4 \sqrt{3} = 12 \sqrt{3} \)

Step 4 - Add the like terms

Finally, add the like terms \(4 \sqrt{3} + 12 \sqrt{3} \): \( 4 \sqrt{3} + 12 \sqrt{3} = 16 \sqrt{3} \)

Key Concepts

Simplification of Square Roots

When simplifying square roots, the goal is to express the square root in its simplest form. This often involves breaking down the number under the square root into its prime factors or finding perfect square factors. For instance, in our problem, we simplified \(\sqrt{12}\) and \(\sqrt{48}\).

Consider \(\sqrt{12}\). We can rewrite 12 as 4 times 3, since 4 is a perfect square. Therefore, \(\sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2 \sqrt{3}\).

Similarly for \(\sqrt{48}\), we can express 48 as 16 times 3, with 16 being a perfect square. Thus, \(\sqrt{48} = \sqrt{16 \cdot 3} = \sqrt{16} \cdot \sqrt{3} = 4 \sqrt{3}\).

By recognizing the factors, we can simplify the square roots to more manageable terms, making further calculations easier.

Like Terms

When simplifying expressions, it's crucial to identify like terms so you can combine them. Like terms are terms that contain the same variables raised to the same power. In the context of square roots, like terms have the same square root component.

In our example, after simplifying the square roots, we ended up with \(2 \sqrt{3}\) and \(4 \sqrt{3}\). When we replaced these in the expression, we got:
[2 \cdot 2 \sqrt{3} + 3 \cdot 4 \sqrt{3}] which simplified further to, \[4 \sqrt{3} + 12 \sqrt{3}\].

Notice that both terms have \(\sqrt{3}\) as a common factor. This commonality means they are like terms and can be added together.

• Combining like terms simplifies the expression and helps in solving equations more easily.
• Always ensure that terms share the exact variable part before adding them together.

Coefficient Multiplication

Coefficients are the numerical parts of terms in an expression. When simplifying expressions involving radicals, correctly multiplying the coefficients is crucial.

In our example, we needed to multiply the coefficients with the simplified square roots. For the expression \(2 \sqrt{12} + 3 \sqrt{48}\), after simplifying the square roots, we had:

\[2 \cdot 2 \sqrt{3} + 3 \cdot 4 \sqrt{3}] \]

Here’s how coefficient multiplication worked:

• First term: \(2 \cdot 2 \sqrt{3} = 4 \sqrt{3}\)
• Second term: \(3 \cdot 4 \sqrt{3} = 12 \sqrt{3}\)

By multiplying the numerical coefficients and then combining like terms, we achieved the final result,
\[4 \sqrt{3} + 12 \sqrt{3} = 16 \sqrt{3}\]. Coefficient multiplication ensures that each term is accurately simplified, making it easier to combine terms later.