Problem 6 - Algebra 2 and Trigonometry

Question

In \(3-41\) , express each product in simplest form. Variables in the radicand with an even index are non-negative. $$ \sqrt{8} \cdot \sqrt{12} $$

Step-by-Step Solution

Answer

The simplest form is \(4\sqrt{6}\).

Step 1: Use Product Property of Radicals

When we have products of square roots, we can use the property \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \). Therefore, \( \sqrt{8} \cdot \sqrt{12} = \sqrt{8 \cdot 12} \).

Step 2: Simplify the Expression Inside the Radical

Multiply the numbers inside the square root: \( 8 \cdot 12 = 96 \). So the expression becomes \( \sqrt{96} \).

Step 3: Factorize the Number Inside the Square Root

Find the prime factorization of 96: \( 96 = 2^5 \cdot 3 \).

Step 4: Simplify the Square Root by Grouping

Since we're dealing with a square root, look for pairs of prime numbers: \( 2^5 \cdot 3 = (2^2) \cdot (2^2) \cdot 2 \cdot 3 \). Each pair of \( 2^2 \) will come out of the square root as a 2, so \( \sqrt{(2^2) \cdot (2^2) \cdot 2 \cdot 3} = 2 \cdot 2 \cdot \sqrt{2 \cdot 3} \).

Step 5: Final Simplification

Calculate \( 2 \cdot 2 = 4 \), and simplify \( \sqrt{2 \cdot 3} = \sqrt{6} \). Thus, \( \sqrt{96} = 4 \sqrt{6} \).

Key Concepts

Product Property of Radicals

The product property of radicals is a handy tool when dealing with the multiplication of square roots or radical expressions. This property states that when you multiply two square roots, you can combine them under a single square root. In mathematical terms, this is expressed as \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \).

• This property saves time and simplifies calculations.
• It groups the multiplication into one neat expression.
• For example, with \( \sqrt{8} \cdot \sqrt{12} \), we can use it to write \( \sqrt{8 \cdot 12} \).

Using this property is the first step in simplifying radical products.

Square Roots

Square roots are numbers that produce a specified quantity when multiplied by itself. For instance, \( \sqrt{9} \) is 3 because \( 3 \times 3 = 9 \). Square roots are used to simplify expressions that involve powers.

• A square root "undoes" the squaring of each number.
• While they seem complex, they simplify calculations greatly.
• In our exercise, \( \sqrt{96} \) needs further simplification for clarity.

Understanding the nature of square roots is essential in simplifying radical expressions.

Prime Factorization

Prime factorization breaks down a number into a product of prime numbers. These prime numbers signal the smallest building blocks of other numbers. For the expression \( 96 \), the prime factorization is \( 2^5 \cdot 3 \).

• Prime factorization helps simplify radicals by organizing terms.
• It's the second step after combining products using the product property.
• We look out for pairs of prime numbers to simplify expressions, especially useful in square roots.

This step is key to unveiling and pairing factors that ease further simplification.

Simplifying Radicals

Simplifying radicals revolves around reducing them to the simplest form possible. We pair off factors under the root to bring them outside as whole numbers. In our example, the prime factorization of \( 96 \) helps us find pairs.

• Each \( 2^2 \) pair forms a single 2 outside the square root.
• Leftovers remain under the square root, such as \( \sqrt{6} \) in simplest form.
• This process turns \( \sqrt{96} \) into the neat \( 4\sqrt{6} \).

Simplifying radicals make them more manageable and easily understood.