Solved problems: Multiplying radical expressions with detailed solutions

📚 Multiplying Radical Expressions: A Comprehensive Guide

Multiplying radical expressions involves combining terms that include roots (like square roots, cube roots, etc.). It builds upon the basic principles of algebra and requires a solid understanding of how to simplify radicals.

📜 History and Background

The concept of radicals dates back to ancient civilizations, with early forms appearing in Babylonian mathematics. The formalization of radical expressions and their operations developed over centuries, becoming an integral part of algebra by the medieval period. The notation we use today evolved through the work of mathematicians in Europe during the Renaissance.

🔑 Key Principles

• 🔍 Product Rule: The product of two radicals with the same index is the radical of the product. Mathematically, $\sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{ab}$.
• 💡 Simplifying Radicals: Before multiplying, simplify each radical expression. This involves factoring out perfect squares (or cubes, etc.) from the radicand (the number under the radical sign).
• 📝 Distributive Property: When multiplying a radical expression by a sum or difference, use the distributive property. For example, $a(\sqrt{b} + \sqrt{c}) = a\sqrt{b} + a\sqrt{c}$.
• 🔢 Combining Like Terms: After multiplying and simplifying, combine any like terms. Like terms have the same radical part.

🧮 Solved Problems

Example 1: Multiplying Simple Square Roots

Multiply $\sqrt{3} \cdot \sqrt{5}$

Solution:

Using the product rule, $\sqrt{3} \cdot \sqrt{5} = \sqrt{3 \cdot 5} = \sqrt{15}$

Example 2: Multiplying Radicals with Coefficients

Multiply $2\sqrt{7} \cdot 3\sqrt{2}$

Solution:

Multiply the coefficients and the radicals separately: $(2 \cdot 3)(\sqrt{7} \cdot \sqrt{2}) = 6\sqrt{14}$

Example 3: Multiplying Radicals with Variables

Multiply $\sqrt{2x} \cdot \sqrt{8x}$

Solution:

$\sqrt{2x} \cdot \sqrt{8x} = \sqrt{16x^2} = 4x$

Example 4: Multiplying a Radical Expression by a Sum

Multiply $\sqrt{2}(\sqrt{3} + \sqrt{5})$

Solution:

Using the distributive property: $\sqrt{2}(\sqrt{3} + \sqrt{5}) = \sqrt{6} + \sqrt{10}$

Example 5: Multiplying Binomial Radical Expressions

Multiply $(\sqrt{2} + \sqrt{3})(\sqrt{5} - \sqrt{2})$

Solution:

Use the FOIL method (First, Outer, Inner, Last): $(\sqrt{2} \cdot \sqrt{5}) + (\sqrt{2} \cdot -\sqrt{2}) + (\sqrt{3} \cdot \sqrt{5}) + (\sqrt{3} \cdot -\sqrt{2}) = \sqrt{10} - 2 + \sqrt{15} - \sqrt{6}$

Example 6: Multiplying Cube Roots

Multiply $\sqrt[3]{4} \cdot \sqrt[3]{2}$

Solution:

$\sqrt[3]{4} \cdot \sqrt[3]{2} = \sqrt[3]{8} = 2$

Example 7: Simplifying Before Multiplying

Multiply $\sqrt{18} \cdot \sqrt{8}$

Solution:

First, simplify the radicals: $\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}$ and $\sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2}$ Then, multiply: $3\sqrt{2} \cdot 2\sqrt{2} = (3 \cdot 2)(\sqrt{2} \cdot \sqrt{2}) = 6 \cdot 2 = 12$

✅ Practice Quiz

Solve the following radical multiplication problems:

1. Multiply $\sqrt{5} \cdot \sqrt{6}$
2. Multiply $3\sqrt{2} \cdot 4\sqrt{3}$
3. Multiply $\sqrt{3x} \cdot \sqrt{12x}$
4. Multiply $\sqrt{3}(\sqrt{2} + \sqrt{7})$
5. Multiply $(\sqrt{5} + \sqrt{2})(\sqrt{3} - \sqrt{2})$
6. Multiply $\sqrt[3]{9} \cdot \sqrt[3]{3}$
7. Multiply $\sqrt{27} \cdot \sqrt{3}$

💡 Conclusion

Multiplying radical expressions involves applying the product rule, simplifying radicals, and using the distributive property. By understanding these principles and practicing regularly, you can master the multiplication of radical expressions. Remember to always simplify your answers and combine like terms where possible.