Problem 74 - Algebra 1: Concepts and Skills

Question

Simplify the radical expression. \(\frac{1}{4} \sqrt{112}\)

Step-by-Step Solution

Answer

\(\sqrt{7}\)

Step 1: Simplify the Numerical Part

The numerical part of the given expression is \(\frac{1}{4}\). This is already in the simplest form.

Step 2: Factorize the Radical Expression

Let's factorize 112. The prime factorization of 112 is \(2^4 \times 7\). Therefore, we write \(\sqrt{112}\) as \(\sqrt{2^4 \times 7}\)

Step 3: Simplify the Radical Expression

We can now simplify \(\sqrt{2^4 \times 7}\) by taking out pairs of 2's from the square root which results in \(2^2\sqrt{7}\). So, the expression \(\sqrt{112}\) simplifies to \(2^2\sqrt{7}\) or \(4\sqrt{7}\)

Step 4: Combine the Numerical and Radical Parts

Now we combine \(\frac{1}{4}\) (from Step 1) and \(4\sqrt{7}\) (from Step 3) using multiplication. \(\frac{1}{4} \times 4\sqrt{7} = \sqrt{7}\)

Key Concepts

Prime Factorization

Prime factorization is a method used to express a number as a product of its prime numbers, which are numbers greater than 1 with no divisors other than 1 and themselves. This process is crucial when simplifying square roots, as it allows us to break down the number under the radical into its simplest components.

For example, to factorize 112, we start by dividing by the smallest prime number, which is 2:

• 112 ÷ 2 = 56
• 56 ÷ 2 = 28
• 28 ÷ 2 = 14
• 14 ÷ 2 = 7

Since 7 is a prime number, we stop there, resulting in the prime factorization: \[112 = 2^4 \times 7\]

This process simplifies the task of working with radical expressions, making subsequent steps more straightforward.

Simplification of Square Roots

Simplifying square roots involves expressing the radical in its simplest form, often using its prime factorization. After factorizing the number inside the square root, we look for pairs of identical factors since the square root of a pair of numbers is a whole number.

In the case of \( \sqrt{112} \), its prime factorization is \( 2^4 \times 7 \). Since \( 2^4 \) equates to two pairs of 2's, these can be simplified as follows:

• The square root of \(2^4\) is \(2^2\), because \(\sqrt{2^4}\) equals \(2^2\).
• Thus, \( \sqrt{2^4 \times 7} = 2^2\sqrt{7}\)
• The expression simplifies further to \( 4\sqrt{7} \) since \(2^2 = 4\).

This step is fundamental in transforming a complex radical into a more manageable expression.

Combination of Numerical and Radical Parts

The final stage of simplifying radical expressions is combining any existing numerical prefactor with the simplified radical part. This step integrates the two distinct components using basic arithmetic operations.

In this exercise, we have the numerical part \( \frac{1}{4} \) and the simplified radical part \(4\sqrt{7}\). To combine them, we multiply these parts:

• Start with \( \frac{1}{4} \times 4\sqrt{7} \).
• The 4s in \( \frac{1}{4} \) and \(4\sqrt{7}\) cancel out, leaving exactly \( \sqrt{7} \).

This shows that often, a simplified radical expression can further declutter once the numerical and radical components are properly combined, showcasing the beauty of how mathematics simplifies complex problems.