Problem 12 - Prealgebra

Question

Simplify each square root, then combine if possible. Assume all variables represent positive numbers. $$\sqrt{12}+\sqrt{27}$$

Step-by-Step Solution

Answer

The expression simplifies to \( 5\sqrt{3} \).

Step 1: Simplify \( \sqrt{12} \)

To simplify \( \sqrt{12} \), start by finding the prime factorization of 12, which is \( 12 = 2^2 \times 3 \). Use the property \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \) to rewrite \( \sqrt{12} \) as \( \sqrt{4 \times 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3} \).

Step 2: Simplify \( \sqrt{27} \)

Find the prime factorization of 27, which is \( 27 = 3^3 \). Use the same square root property to rewrite \( \sqrt{27} \) as \( \sqrt{3^2 \times 3} = \sqrt{9} \cdot \sqrt{3} = 3\sqrt{3} \).

Step 3: Combine the simplified expressions

Now that both square roots are simplified, combine them: \( 2\sqrt{3} + 3\sqrt{3} \). Since both terms have the same radical part \( \sqrt{3} \), you can add the coefficients: \( 2 + 3 = 5 \). Thus, the expression simplifies to \( 5\sqrt{3} \).

Key Concepts

Radical Expressions

Radical expressions can seem intimidating at first, but they are simply expressions that involve roots, such as square roots. The square root, often denoted by the radical sign \(\sqrt{}\), is the most common type of radical expression. When you see \(\sqrt{\cdot}\), it indicates you need to find what number results in the original value when squared. For example, \(\sqrt{9} = 3\) because \(3^2 = 9\).
Radicals can include coefficients, such as \(2\sqrt{3}\), which means two times the square root of 3. Simplifying radical expressions often involves identifying perfect squares within the radicand — the number inside the radical sign.

• Always check if the radicand can be broken down into smaller factors that include a perfect square.
• This makes simplification possible, allowing you to write the expression in simpler terms.
• For instance, \(\sqrt{12}\) simplifies to \(2\sqrt{3}\) due to the perfect square factor of 4 in 12.

Prime Factorization

Prime factorization is a valuable tool when it comes to simplifying square roots. It involves breaking a number down into the product of prime numbers — numbers that are divisible only by 1 and themselves. For example, the prime factorization of 12 is \(2^2 \times 3\). This step is crucial because it helps identify whether any pairs of factors — which are perfect squares — exist within the radicand.
Identifying these pairs lets you take them out of the radical as whole numbers, simplifying the expression. For example:

• Step 1: Identify the numbers you can multiply to get the original number (e.g., \(12 = 2 \times 6\)).
• Step 2: Further break these numbers down into their prime numbers (e.g., \(6 = 2 \times 3\)).
• Step 3: Express the original number entirely in terms of prime factors (e.g., \(12 = 2^2 \times 3\)).

Using prime factorization can turn something complicated into something much easier to handle.

Combining Like Terms

Combining like terms is a fundamental algebraic principle that can simplify expressions. It's handy when you're adding or subtracting radical expressions with the same radicand. Terms are considered 'like' if they have identical variable parts or, in this context, the same radical parts. Consider the expression \(2\sqrt{3} + 3\sqrt{3}\). Both terms have the same radical part \(\sqrt{3}\).
When combining these like terms:

• Keep the radical part unchanged.
• Add or subtract only the coefficients.
• Here, you calculate \(2 + 3\), which equals 5, to get the final simplified result of \(5\sqrt{3}\).

This principle streamlines calculations and enables you to simplify expressions to their core components swiftly.