Problem 38 - Algebra 2

Question

Simplify. \(\sqrt{12}+\sqrt{48}-\sqrt{27}\)

Step-by-Step Solution

Answer

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

Step 1: Express Each Term in Prime Factors

First, express each of the numbers inside the square roots as a product of their prime factors. For 12, this is \(12 = 2^2 \times 3\). For 48, this is \(48 = 2^4 \times 3\). For 27, this is \(27 = 3^3\).

Step 2: Simplify Each Square Root

Convert the prime factorization into square roots and simplify. For \(\sqrt{12}\), split into \(\sqrt{2^2 \times 3} = \sqrt{2^2} \times \sqrt{3} = 2\sqrt{3}\). For \(\sqrt{48}\), split into \(\sqrt{2^4 \times 3} = \sqrt{2^4} \times \sqrt{3} = 4\sqrt{3}\). For \(\sqrt{27}\), split into \(\sqrt{3^3} = \sqrt{3^2 \times 3} = 3\sqrt{3}\).

Step 3: Combine Like Terms

Add and subtract the simplified forms: \(2\sqrt{3} + 4\sqrt{3} - 3\sqrt{3}\). These are like terms because they all involve \(\sqrt{3}\).

Step 4: Perform the Arithmetic

Combine the coefficients of \(\sqrt{3}\): \(2 + 4 - 3 = 3\). Therefore, the expression simplifies to \(3\sqrt{3}\).

Key Concepts

Prime Factorization

Prime factorization is the process of breaking down a number into its smallest prime numbers that multiply together to give the original number. This is the first step in simplifying radical expressions. By identifying the prime factors, you can simplify square roots. Let's take a quick look at how you would perform prime factorization on some numbers:

• For the number 12, you repeatedly divide by prime numbers like 2 or 3. This gives us, 12 divided by 2 is 6, and again dividing 6 by 2 gives 3. In the end, 12 is expressed as \(2 \times 2 \times 3\) or as powers, \(2^2 \times 3\).
• Similarly, for the number 48, divide by 2 to get 24, 24 by 2 to get 12, 12 by 2 to get 6, and 6 by 2 to get 3, thus resulting in \(2^4 \times 3\).
• For 27, you divide by 3 to get 9, and 9 by 3 to get 3, expressed as \(3^3\).

By performing prime factorization, you lay the groundwork for simplifying square roots, which is critical in working with radical expressions.

Square Roots

Square roots help in undoing the power of 2. Simply put, the square root of a number is a value that, when multiplied by itself, gives the initial number. Understanding square roots is important in simplifying radicals. In the context of simplifying radicals:

• You look at the prime factors obtained. For example, from the expression \(\sqrt{12}\), we know the factors are \(2^2 \times 3\).
• Take square roots of perfect squares. In \(\sqrt{2^2 \times 3}\), \(\sqrt{2^2}\) becomes 2 and is taken outside the root, resulting in \(2\sqrt{3}\).
• This method applies to \(\sqrt{48}\) and \(\sqrt{27}\) too, breaking down to \(4\sqrt{3}\) and \(3\sqrt{3}\) respectively.

It’s helpful to remember that perfect squares simplify neatly, and exploring these values simplifies the entire expression efficiently.

Simplify Radicals

Simplifying radicals means reducing a radical to its simplest form. This process involves using prime factorization and square root knowledge to combine terms effectively.Here's how you simplify a radical step by step:

• First, perform prime factorization of the number under the root.
• Then, pull out pairs of prime factors out of the square root to the left as their square root adjustments.
• Lastly, combine any like terms. Like in the exercise, after simplifying, we had \(2\sqrt{3}\), \(4\sqrt{3}\), and \(3\sqrt{3}\).

The last step involves arithmetic with these terms: adding and subtracting their coefficients. Here, \(2 + 4 - 3\) gives us a final simplified form of \(3\sqrt{3}\). Simplifying radicals not only makes expressions compact and clear but also aids in better understanding relationships between numbers, which is valuable for solving various mathematical problems.