Problem 8 - 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. $$ 3 \sqrt{2} \cdot \sqrt{10} $$

Step-by-Step Solution

Answer

The expression simplifies to \(6\sqrt{5}\).

Step 1: Understand the Problem

We need to multiply two square roots: \(3\sqrt{2}\) and \(\sqrt{10}\). Our goal is to express the result as a single term in its simplest form.

Step 2: Apply the Product Property of Square Roots

According to the product property of square roots, \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\). Here, we can multiply the numbers under the square roots: \(\sqrt{2} \cdot \sqrt{10} = \sqrt{20}\).

Step 3: Simplify the Expression

Next, simplify \(3\sqrt{20}\). Factor \(20\) as \(4 \times 5\) to get: \(\sqrt{20} = \sqrt{4} \cdot \sqrt{5} = 2\sqrt{5}\) because \(\sqrt{4} = 2\). Now, multiply by 3: \(3 \cdot 2\sqrt{5} = 6\sqrt{5}\).

Step 4: Verify the Simplified Form

The expression \(6\sqrt{5}\) cannot be simplified further under the given problem constraints. Therefore, the simplest form of the product is confirmed.

Key Concepts

Product of Square Roots

Understanding the product of square roots is crucial when simplifying expressions involving radicals. When given two square roots to multiply, we can simplify the operation using the product property of square roots. This handy mathematical property states that the square root of a product is equal to the product of the square roots. In symbolic form, it is expressed as follows: - If you have two numbers, say 'a' and 'b', the product property allows you to calculate: \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \).Using this property, computations become much more manageable. For example, with our example of \(3\sqrt{2} \cdot \sqrt{10} \), one can combine \(\sqrt{2}\) and \(\sqrt{10}\) into a single radical: \( \sqrt{20} \). It simplifies not only the multiplication itself but also prepares the expression for further simplification steps.

Simplification Process

The simplification process of square roots involves reducing the radicals to their simplest form by factoring them. After using the product property to merge square roots, the next step is simplification:1. **Factor the radicand**: Here, find a pair of numbers where one is a perfect square. In \(\sqrt{20}\), notice that 20 can be factored into \(4\times 5\), with 4 being a perfect square.2. **Simplify the square root**: Once factored, separate the square root \(\sqrt{20} = \sqrt{4} \cdot \sqrt{5} = 2\sqrt{5} \). Since \(\sqrt{4}\) is simply 2, this separates terms for easier manipulation.3. **Combine results**: Finally, incorporate any coefficients from the initial expression. Multiply back, in our example, the 3 outside the radical combines with 2 inside: \(3 \cdot 2\sqrt{5} = 6\sqrt{5}\).The process requires careful factorization and simple arithmetic to ensure the radical is in its lowest terms.

Square Root Properties

Square root properties are foundational rules that help us manipulate and simplify expressions involving roots. Understanding these properties allows for smoother transitions between different forms of expressions:- **Product Property**: As mentioned, this allows combining roots under a single radical: \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \).- **Quotient Property**: This property allows simplification of division inside a square root: \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\), provided that \(b eq 0\).- **Radical Simplification**: If the radicand contains a perfect square, you can simplify it directly. This is essential during the factorization step.By mastering these properties, tackling any expression involving square roots becomes easier. They guide each step of solving, providing multiple angles to approach and simplify the given problems effectively.