Problem 23 - Algebra for College Students

Question

Change each radical to simplest radical form. \(\sqrt{32}\)

Step-by-Step Solution

Answer

\(\sqrt{32} = 4\sqrt{2}\).

Step 1: Identify the Prime Factors

To simplify the square root, start by identifying the prime factors of the number under the radical. Here, we need the prime factors of 32. The prime factorization of 32 is 2 * 2 * 2 * 2 * 2 or \(2^5\).

Step 2: Group the Prime Factors into Pairs

In square root simplification, we pair the factors because \(\sqrt{x^2} = x\), meaning a pair of the same numbers under a square root can be simplified to one of those numbers. From \(2^5\), we have two pairs of 2s: \((2 * 2), (2 * 2)\), and an extra factor of 2.

Step 3: Simplify the Radical

We rewrite the square root expression using the pairs. We know \((2 * 2) = 4\), so \((2^2) = 4\) turns into \(\sqrt{4} = 2\). Therefore, each pair simplifies to 2. Since we have two pairs, we bring out 2 * 2, making it \(4\). We are left with one 2 inside the square root: \(\sqrt{32} = 4\sqrt{2}\).

Step 4: Write the Final Simplified Form

Combine the simplified numbers outside the square root with what remains under the radical to get the final answer. Thus, \(\sqrt{32} = 4\sqrt{2}\).

Key Concepts

Prime Factorization

Prime factorization is a method used to express a number as a product of its prime numbers. It is essential in simplifying radicals. Prime numbers are numbers greater than 1 that only have two factors: 1 and themselves.
For instance, when we want to simplify the square root of 32, we first look for its prime factorization.

• Start with the number 32.
• Divide by the smallest prime number, which is 2:
  • 32 divided by 2 is 16.
  • 16 divided by 2 is 8.
  • 8 divided by 2 is 4.
  • 4 divided by 2 is 2.
  • 2 divided by 2 is 1.
• This gives us the prime factors: 2 * 2 * 2 * 2 * 2, or in exponential form, \(2^5\).

This step forms the basis for simplifying the radical, providing a clear path to recognizing and grouping pairs of prime numbers.

Square Roots

Square roots are an essential element of mathematics, representing a value that, when multiplied by itself, gives the original number under the radical. The symbol \(\sqrt{}\) represents the square root.
Simplifying square roots involves identifying pairs of identical factors derived from prime factorization, as any pair can be simplified to a single number.
For the square root of 32, prime factorization gives \(2^5\).

• Let's look for pairs in \(2^5\):
  • There are two pairs of 2s: \((2 \times 2), (2 \times 2)\).
  • This leaves one 2 unpaired.

Each pair can be pulled out of the square root as a single 2 because \(\sqrt{2 \times 2} = 2\). Combining these results gives us the simplified radical expression. This step is crucial to moving forward in simplifying radicals efficiently.

Radical Expressions

Radical expressions encompass expressions that include a root, such as square roots, cube roots, etc. Simplifying these expressions requires breaking down components into simpler forms that are easier to work with.
In our example with \(\sqrt{32}\), the prime factorization process and the identification of square pairs created a situation where we can transform more complex expressions into simpler ones.

• Pulled-out pairs from the square root turns two pairs of 2s into \(2 \times 2\) or \(4\).
• The leftover solitary factor 2 remains under the square root.

Thus, \(\sqrt{32}\) simplifies to \(4\sqrt{2}\). By reducing the radicals, we achieve clarity and simplicity, making the expression easier to understand, compare, or compute with other radical expressions. Ensuring you can handle such simplifications empowers effective problem-solving in mathematics.