Problem 65 - Algebra 1

Question

Simplify the radical expression. $$\frac{1}{2} \sqrt{28}$$

Step-by-Step Solution

Answer

So, the simplified form of the given radical expression \( \frac{1}{2} \sqrt{28} \) is \( \sqrt{7} \).

Step 1: Perform Prime Factorization of the Number Under the Root

We begin with the prime factorization of 28, which is 2 * 2 * 7.

Step 2: Use the Property of Radicals to Separate Them

Using the property that the square root of a product equals the product of square roots, we can now separate the numbers under the root. Thus, we get \( \frac{1}{2} \sqrt{2*2*7} \) = \( \frac{1}{2} \sqrt{2*2} \) * \( \sqrt{7} \)

Step 3: Simplify the Expression

Squares under roots can be taken out as they are. We know that \( \sqrt{2*2} \) gives us 2. So we have \( \frac{1}{2} \) * 2* \( \sqrt{7} \), simplifying this gives us \( \sqrt{7} \).

Key Concepts

Prime Factorization

Prime factorization is the process of breaking down a composite number into its prime factors. A prime number is a number that has only two distinct positive divisors: 1 and itself. For example, 2, 3, 5, and 7 are prime numbers, while 28 can be divided further.

• To perform prime factorization on 28, we start dividing by the smallest prime number, which is 2.
• Since 28 is even, it can be divided by 2, giving us 14.
• We continue factoring 14 by 2, resulting in 7.
• Finally, 7 is already a prime number.

So, the prime factorization of 28 is given as 2 * 2 * 7. This method reveals the building blocks of the number 28, which can later help in simplifying expressions involving square roots.

Simplifying Radicals

Simplifying radicals involves reducing the expression under the root symbol to its simplest form. When we simplify a radical expression, we're looking for perfect squares that can "escape" from under the root.

• Begin with the prime factorized form from earlier. Here, it's 2 * 2 * 7 under the square root sign.
• Use the factor of 2 * 2 to simplify, as the square root of 2 squared (\( \sqrt{2^2} \)) becomes 2.
• By separating the square root of the product: \( \sqrt{2 * 2 * 7} = \sqrt{2 * 2} * \sqrt{7} \)
• Hence, simplifying gives us \( 2 \sqrt{7} \).

This process transforms an initially complex radical expression into its most reduced and simplest form, making it easier to understand and work with.

Properties of Radicals

Understanding properties of radicals is crucial when dealing with expressions involving roots. These properties are powerful tools for simplifying expressions and solving radical equations.

• Product Property: The square root of a product is the product of the square roots, as seen when simplifying \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \).
• Perfect Squares: A perfect square under a radical can be simplified directly to its square root, such as \( \sqrt{4} = 2 \).
• Coefficient Multiplication: When a coefficient is present inside and outside the radical, they can be multiplied together after simplifying the radical, as evidenced by the simplification in our example where \( \frac{1}{2} \) was first multiplied by the simplified root.

These properties not only assist in simplifying expressions but aid in understanding how to manipulate and transform equations involving radicals effectively.