Problem 3 - Algebra 2 and Trigonometry

Question

In \(3-38\) , write each radical in simplest radical form. Variables in the radicand of an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ \sqrt{12} $$

Step-by-Step Solution

Answer

\(\sqrt{12} = 2\sqrt{3}\)

Step 1: Identify the Radicand

In the given problem, the radicand is 12. We need to express this as a product of its factors to simplify the square root.

Step 2: Find Prime Factorization

Determine the prime factors of 12. The number 12 can be expressed as the product of the primes: \(12 = 2 \times 2 \times 3\), or equivalently \(2^2 \times 3\).

Step 3: Simplify the Radical

Rewrite the square root using its factorization: \(\sqrt{12} = \sqrt{2^2 \times 3}\). The square root of a product is the product of the square roots, so \(\sqrt{12} = \sqrt{2^2} \times \sqrt{3}\).

Step 4: Extract Perfect Squares

Since \(\sqrt{2^2}\) is a perfect square, simplify it to 2. Therefore, the expression becomes \(2 \times \sqrt{3}\).

Step 5: Write in Simplest Radical Form

The simplest radical form of \(\sqrt{12}\) is \(2\sqrt{3}\).

Key Concepts

Prime Factorization

To simplify a radical, such as \(\sqrt{12}\), it's important to start with prime factorization. Prime factorization is the process of breaking down a number into its smallest prime number factors. These factors are the building blocks of the number, as primes have no divisors other than 1 and themselves.

• For 12, begin by dividing by the smallest prime, 2, which leaves 6.
• Continue with 6, dividing again by 2, leaving 3, which is also a prime number.

Hence, the prime factorization of 12 is \(2 \times 2 \times 3\) or \(2^2 \times 3\). Knowing this allows us to simplify the radical expression by identifying and extracting perfect square factors.

Perfect Squares

Identifying perfect squares is a crucial step in simplifying radicals. A perfect square is a number that has a whole number as its square root. When simplifying \(\sqrt{12}\), we use its prime factors to spot any perfect squares.

• In \(12 = 2^2 \times 3\), the term \(2^2\) is a perfect square because it equals 4, and \(\sqrt{4} = 2\).

Extracting this perfect square simplifies the expression: \(\sqrt{12} = \sqrt{2^2 \times 3} = \sqrt{2^2} \times \sqrt{3}\). This process of extracting and simplifying via perfect squares is key to finding the simplest form of a radical.

Simplest Form

The simplest form of a radical occurs when all possible perfect square factors have been extracted from under the radical sign. This form is easier to understand and work with. After identifying \(2^2\) as a perfect square in \(\sqrt{12}\), it can be simplified to 2.

• Thus, \(\sqrt{12} = 2 \times \sqrt{3}\).
• This simplification removes all perfect square factors from the radicand, leaving no factors other than 1 that can be removed, ensuring it's in simplest form.

Writing radicals in simplest form is not only a standard of simplification but also brings clarity and precision to mathematical expressions.

Radicals

Radicals, symbolized by the square root sign \(\sqrt{\cdot}\), represent the original number under consideration, called the radicand. Radicals come into play when expressing and solving equations involving roots, especially square roots in algebra.

• For \(\sqrt{12}\), the radicand is 12.
• By managing radicals, such as the example of simplifying \(\sqrt{2^2 \times 3}\) to \(2\sqrt{3}\), we make equations less complex and more understandable.

This manipulation of radicals through factorization and simplification is a fundamental skill in math, bridging basic arithmetic and algebraic concepts.