Problem 34 - 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. $$ \sqrt{128 x^{2}} $$

Step-by-Step Solution

Answer

The simplest radical form is \(8x\sqrt{2}\).

Step 1: Identify Prime Factors

First, we need to find the prime factorization of 128. Divide 128 by 2 repeatedly: 128 / 2 = 64, 64 / 2 = 32, 32 / 2 = 16, 16 / 2 = 8, 8 / 2 = 4, 4 / 2 = 2, 2 / 2 = 1. Therefore, the prime factorization of 128 is \(2^7\).

Step 2: Simplify Radicand

Rewrite the radicand using the prime factors: \(\sqrt{128x^2} = \sqrt{2^7 \cdot x^2}\).

Step 3: Apply Square Root Property

To simplify \(\sqrt{2^7 \cdot x^2}\), use the property \(\sqrt{a^2} = a\): \(\sqrt{2^7} \cdot \sqrt{x^2} = \sqrt{(2^3)^2 \cdot 2^1} \cdot x\).This gives us \(2^3 \cdot x \cdot \sqrt{2} = 8x \cdot \sqrt{2}\).

Step 4: Combine and Simplify

Combine the results to form the simplest radical form: \(8x\sqrt{2}\).

Key Concepts

Prime Factorization

Prime factorization is the process of breaking down a number into its basic building blocks, which are prime numbers. This is crucial when simplifying radicals since it helps reveal the structure of the number under the radical sign. For the number 128, you divide by 2, the smallest prime number, since 128 is even. Continuing to divide by 2 gives:

• 128 divided by 2 equals 64,
• 64 divided by 2 equals 32,
• 32 divided by 2 equals 16,
• 16 divided by 2 equals 8,
• 8 divided by 2 equals 4,
• 4 divided by 2 equals 2,
• 2 divided by 2 equals 1.

After these steps, 128 is expressed completely using only the prime number 2, shown as \(2^7\). Understanding this helps us to later apply square root properties more effectively and to simplify complex values under the radical.

Square Root Property

The square root property is a useful tool in simplifying radicals. It states that the square root of a square is its base, symbolically written as \(\sqrt{a^2} = a\). This property is particularly helpful when dealing with even powers of numbers, as it allows for easy simplification. In our example, we use the property to simplify terms with perfect squares under the radical.
For \(\sqrt{128x^2}\), you can separate the terms to handle each independently:

• The 128, using its prime factorization \(2^7\), can be rewritten as \(\sqrt{(2^3)^2 \cdot 2}\).
• The \(x^2\) can be simplified directly since \(\sqrt{x^2} = x\).

Applying the square root property to \( (2^3)^2 \) gives \(2^3 = 8\). This transforms \(\sqrt{128x^2}\) to \(8x \cdot \sqrt{2}\), where each piece under the radical has been simplified using square root properties.

Simplest Radical Form

The simplest radical form is the most simplified version of a radical expression, where no perfect square factors remain under the radical and all possible factors have been simplified out. In our example \(\sqrt{128x^2}\), we broke it down into simpler components using prime factorization and square root properties.
The steps yielded \(8x\sqrt{2}\) as the simplest radical form. Here's why:

• All instances of perfect squares are simplified, as seen with the \((2^3)^2\) portion becoming \(8\).
• The \(x^2\) is simplified to \(x\) given that \(\sqrt{x^2} = x\).
• No further simplification is possible for \(\sqrt{2}\), which remains as is in the final expression.

The simplest radical form expresses a radical in the most efficient and clear way possible, making it a fundamental part of simplifying radical expressions.