Problem 37 - Algebra for College Students

Question

Change each radical to simplest radical form. \(-\frac{5}{6} \sqrt{28}\)

Step-by-Step Solution

Answer

The simplest radical form is \(-\frac{5}{3} \sqrt{7}\).

Step 1: Simplify the Radicand

Let's begin by simplifying the radical. The expression given is \( \sqrt{28} \). First, we need to factor 28 into its prime factors: \( 28 = 2^2 \times 7 \). From this factorization, we can take the square root of \( 2^2 \), which is 2. Therefore, \( \sqrt{28} = \sqrt{4 \times 7} \) = \( \sqrt{4} \times \sqrt{7} = 2\sqrt{7} \).

Step 2: Apply the Simplified Radical

Now that we have simplified \( \sqrt{28} \) to \( 2\sqrt{7} \), we substitute this back into the original expression: \[ -\frac{5}{6} \sqrt{28} = -\frac{5}{6} \times 2\sqrt{7} \].

Step 3: Simplify the Expression

Now, multiply \( -\frac{5}{6} \) by \( 2\sqrt{7} \). First, multiply the numerical parts: \( -\frac{5}{6} \times 2 = -\frac{10}{6} \). Simplify \( -\frac{10}{6} \) to \( -\frac{5}{3} \). So the expression becomes \( -\frac{5}{3} \sqrt{7} \).

Key Concepts

Prime Factorization

Prime factorization is a useful technique when working with radical expressions. It involves breaking down a number into its prime factors, which are numbers that cannot be divided further except by 1 and themselves. Understanding the prime factors of a number helps us simplify radical expressions more effectively. For instance, in our example where we had the number 28 under the square root, the prime factorization of 28 is done by finding the smallest prime number that divides 28. We find that:

• 28 ÷ 2 = 14
• 14 ÷ 2 = 7
• Since 7 is also a prime number, our factorization stops here.

Therefore, 28 can be expressed as the product of its prime factors: \(28 = 2^2 \times 7\). By recognizing these factors, we can simplify the expression under the square root by taking out perfect squares, which is a fundamental step in simplifying radicals.

Radical Expressions

Radical expressions are mathematical expressions that contain a root symbol, such as a square root. Working with them, especially with square roots, can seem challenging at first, but they become manageable with practice and understanding. The key to simplifying radical expressions is to determine if the number or expression inside the root can be broken down into factors that are perfect squares or cubes (for cube roots, and so on). Take our example, \(\sqrt{28}\):

• Using the prime factorization, we expressed 28 as \(2^2 \times 7\).
• A paired factor, like \(2^2\), can come out of the square root as a single 2, since \(\sqrt{2^2} = 2\).

Consequently, our expression simplifies to \(2\sqrt{7}\). The expression is now in a form that's easier to handle in both multiplication or further algebraic operations.

Simplest Form

Expressing a radical in its simplest form is an important step in solving and working with these expressions. The simplest form occurs when no factor inside the radical can be divided out further as a perfect square.Returning to our exercise, once we simplified \(\sqrt{28}\) to \(2\sqrt{7}\), we were almost at the simplest form. We simplified the coefficient of the expression step-by-step:

• Initially, we had \(-\frac{5}{6} \times 2\sqrt{7}\).
• Multiplying the numeric parts, \(-\frac{5}{6} \times 2\) gives \(-\frac{10}{6}\), which simplifies to \(-\frac{5}{3}\).
• This results in the simplest form of the expression: \(-\frac{5}{3}\sqrt{7}\).

What makes this important is that the expression is now fully simplified and ready for any further mathematical operation, without any additional simplifications needed. Recognizing when an expression is in its simplest form ensures clarity and correctness in solving mathematical problems.