The exact value of √10 × √15 is 5√6.
Understanding how to multiply and simplify radical expressions involves applying basic properties of square roots and prime factorization. When multiplying two square roots, you can combine the numbers inside the radical (radicands) under a single square root sign.
Understanding Radical Expressions
Radical expressions, often involving square roots, can be multiplied using a simple property:
- Product Property of Square Roots: For any non-negative real numbers 'a' and 'b', the product of their square roots is equal to the square root of their product.
- √a × √b = √(a × b)
This property allows us to combine multiple square roots into a single, potentially simpler, radical.
Step-by-Step Calculation of √10 × √15
Let's break down the process to arrive at the solution.
Step 1: Combine the Radicands
Using the product property of square roots, multiply the numbers inside the radicals:
- √10 × √15 = √(10 × 15)
- = √150
Step 2: Prime Factorization
To simplify √150, we need to find the prime factors of 150. This helps in identifying any perfect square factors that can be extracted from the radical.
- Start by breaking down 150 into its smallest prime components:
- 150 = 10 × 15
- 10 = 2 × 5
- 15 = 3 × 5
- So, the prime factorization of 150 is 2 × 3 × 5 × 5, or 2 × 3 × 5².
Step 3: Simplify the Radical
Now, substitute the prime factors back into the radical expression and look for pairs of identical factors (which represent perfect squares).
- √150 = √(2 × 3 × 5²)
- Since 5² is a perfect square (25), its square root is 5. We can take this 5 outside the radical.
- The remaining factors inside the radical are 2 and 3.
- √150 = √(5²) × √(2 × 3)
- = 5 × √6
- = 5√6
This result, 5√6, is the most simplified form of the expression.
Summary of Simplification
| Step | Description | Calculation |
|---|---|---|
| Multiply Radicals | Combine numbers under one root | √10 × √15 = √150 |
| Prime Factorization | Find prime factors of the radicand | 150 = 2 × 3 × 5² |
| Extract Perfect Squares | Pull out perfect square roots | √150 = √(5² × 2 × 3) = 5√6 |
Key Takeaways for Simplifying Radicals
- Combine first: Always multiply numbers inside the square roots before attempting to simplify, if applicable.
- Prime Factorize: Breaking down the radicand into its prime factors is crucial for identifying perfect square components.
- Look for Pairs: For square roots, identify pairs of identical prime factors. Each pair can be pulled out of the radical as a single factor.
- Multiply Outside, Multiply Inside: Any numbers extracted from the radical are multiplied with each other (and with any existing numbers outside the radical). Any remaining factors inside the radical are multiplied together.
