Problem 85 - Algebra 1: Concepts and Skills

Question

Write the radical expression in simplest form. $$ \frac{1}{2} \sqrt{32} \cdot \sqrt{2} $$

Step-by-Step Solution

Answer

The simplest form of the given radical expression is \(4\).

Step 1: Simplify the Square Roots

Simplify the square roots by factoring out the squares. The square root of \(32\) can be written as \(\sqrt{16 \times 2}\) which simplifies to \(4\sqrt{2}\). The expression then becomes: \(\frac{1}{2} \cdot 4\sqrt{2} \cdot \sqrt{2}\)

Step 2: Simplify the Multiplication

By simplifying the multiplication, it will be \(\frac{1}{2} \cdot 4\sqrt{2} \cdot \sqrt{2} = 2\sqrt{2} \cdot \sqrt{2} = 2\cdot 2 = 4\). Thus, the expression simplifies to \(4\)

Step 3: Express in Simplest Form

The simplified form of the expression is \(4\)

Key Concepts

Square Roots

Square roots are a fundamental concept in mathematics. They represent a number which, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because 3 times 3 equals 9. Square roots are symbolized by the radical sign \(\sqrt{}\). When simplifying square roots, like \(\sqrt{32}\), it is important to look for perfect squares. In this case, \(\sqrt{32}\) can be decomposed into \(\sqrt{16} \cdot \sqrt{2}\), where \(\sqrt{16} = 4\). Thus, \(\sqrt{32} = 4\sqrt{2}\).

• Perfect squares nearest to 32 are 16 (4²) and 25 (5²).
• Look for factors that are also perfect squares to simplify.

Remember, simplifying square roots involves finding these perfect square factors.

Multiplication of Radicals

When multiplying radicals, or square roots, there's a key property to remember: the product rule of radicals. This property states that \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\). By using this rule, you can combine and simplify radical expressions. In our example, \(4\sqrt{2} \cdot \sqrt{2}\) can be simplified by the product rule. Combining the radicals, we have \(4 \cdot \sqrt{2 \cdot 2} = 4 \cdot \sqrt{4}\). Since \(\sqrt{4} = 2\), the expression becomes \(4 \cdot 2 = 8\). To ensure accurate multiplication results:

• Always check the radicals can be combined by using the product rule.
• Simplify all possible roots after multiplication for a cleaner result.

Multiplying radicals effectively can make simplifying the expression straightforward and produce a simpler form.

Factoring Squares

Factoring squares is an efficient technique to simplify radical expressions. This process involves breaking down a number into its factor components, which include perfect square factors. Factoring squares helps in identifying those components which can be "taken out" from under the radical sign. Consider \(\sqrt{32}\), broken down into its factors, it becomes \(\sqrt{16 \times 2}\), where 16 is a perfect square. Factoring the square enables the calculation \(\sqrt{16} = 4\), eventually simplifying \(\sqrt{32}\) to \(4\sqrt{2}\). Key points to remember:

• Identify largest perfect square factors for easier simplification of radicals.
• Re-factor complex numbers into products of smaller perfect squares.

This practice is not only useful for simplifying individual roots but also for simplifying the broader expressions containing radicals.