Problem 5 - 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. $$ \sqrt{32} $$

Step-by-Step Solution

Answer

The simplest radical form of \( \sqrt{32} \) is \( 4 \sqrt{2} \).

Step 1: Identify the Radicand

In this problem, we are given the expression \( \sqrt{32} \). The radicand here is 32, which is the number inside the square root that we need to simplify.

Step 2: Prime Factorization

Find the prime factors of 32 by dividing it by the smallest prime number repeatedly: \( 32 \div 2 = 16 \), \( 16 \div 2 = 8 \), \( 8 \div 2 = 4 \), \( 4 \div 2 = 2 \), and \( 2 \div 2 = 1 \). So, the prime factorization of 32 is \( 2^5 \).

Step 3: Apply the Square Root Property

The square root of a number is the product of the square roots of its prime factors: \( \sqrt{32} = \sqrt{2^5} \). This can be broken down further using properties of exponents and square roots: \( \sqrt{32} = \sqrt{2 \cdot 2 \cdot 2 \cdot 2 \cdot 2} = \sqrt{(2^2)^2 \times 2} \).

Step 4: Simplify Using Perfect Squares

Since \( 2^2 \) is a perfect square, we can take it out of the square root as \( 2 \). We do this for each pair: \( \sqrt{(2^2)^2 \times 2} = 2^2 \sqrt{2} = 4 \sqrt{2} \).

Step 5: Verify the Simplified Form

Finally, check if the expression can be simplified further. \( 4 \sqrt{2} \) is in simplest radical form as there are no more perfect squares in the radicand 2.

Key Concepts

Prime Factorization

Prime factorization is the process of decomposing a number into its fundamental building blocks: prime numbers. Think of prime numbers as the "atoms" of the mathematical world that cannot be broken down any further. For instance, when simplifying radicals like \( \sqrt{32} \), the first step is to break down 32 into prime factors.
Start with the smallest prime number, which is 2, and continuously divide until you can't anymore. Here’s how it's done for 32:

• Divide 32 by 2: \( 32 \div 2 = 16 \)
• Divide 16 by 2: \( 16 \div 2 = 8 \)
• Divide 8 by 2: \( 8 \div 2 = 4 \)
• Divide 4 by 2: \( 4 \div 2 = 2 \)
• Finally, divide 2 by 2: \( 2 \div 2 = 1 \)

Once completed, you'll find that the prime factorization of 32 is \( 2^5 \). With all the factors individually being prime, the stage is set for further simplification.

Square Root Property

The square root property is a fundamental concept that involves the simplification of square roots. When looking at a radical such as \( \sqrt{2^5} \), the square root property allows us to simplify it based on the factors.
A key principle here is that the square root of a product is the product of the square roots: \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \). Utilizing this, we can break down \( \sqrt{32} \) as follows:

• Express 32 as \( 2^5 \)
• Recognize \( 2^5 \) as \( 2 \times 2 \times 2 \times 2 \times 2 \)
• Group these elements to form perfect squares: \( (2^2)^2 \times 2 \)
• Apply the square root: \( \sqrt{(2^2)^2} \times \sqrt{2} \)
• Since \( \sqrt{(2^2)^2} = 2^2 \), simplify to \( 4 \sqrt{2} \)

This property, paired with prime factorization, forms the backbone for simplifying radicals efficiently.

Perfect Squares

Perfect squares are numbers whose square roots are integers. Recognizing them is crucial for simplifying radicals. In the context of our example \( \sqrt{32} \), identifying perfect squares among the factors of 32 allows us to simplify more effectively.
Perfect squares result from squaring a whole number. For instance, 1, 4, 9, 16, and so on are perfect squares because they are 1², 2², 3², 4², respectively.
In \( \sqrt{32} = \sqrt{2^5} \), we focus on the largest possible perfect squares:

• Recognize that \( 2^4 = (2^2)^2 = 4^2 = 16 \), a perfect square
• Express \( 32 \) as a product involving perfect squares \( (2^2)^2 \times 2 \)
• Applying \( \sqrt{16} = 4 \), we get: \( 4 \sqrt{2} \)

By knowing and utilizing perfect squares, you can simplify radicals without trial and error, making processes like these much faster and increasing your confidence in handling radical expressions.