How to Simplify Expressions with Square Roots: A Step-by-Step Guide
Quick Answer
To simplify the expression $- ext{sqrt}{5}(-10 - ext{sqrt}{3})$, distribute $- ext{sqrt}{5}$ to both terms. The final answer is $10 ext{sqrt}{5} + ext{sqrt}{15}$.
Simplifying expressions involving square roots can seem challenging at first, but with a clear method, you can tackle them step by step. Let's take a closer look at the expression $- ext{sqrt}{5}(-10 - ext{sqrt}{3})$.
### Step 1: Understand the Problem
The expression consists of two parts inside parentheses: $-10$ and $- ext{sqrt}{3}$. We need to distribute the $- ext{sqrt}{5}$ to both parts. Distribution is a fundamental property in algebra that allows us to multiply a single term by each term inside parentheses.
### Step 2: Distributing $- ext{sqrt}{5}$
Let’s break down the distribution:
1. **First Term**: Multiply $- ext{sqrt}{5}$ by $-10$:
- $- ext{sqrt}{5} imes -10 = 10 ext{sqrt}{5}$ (Remember, a negative times a negative equals a positive.)
2. **Second Term**: Multiply $- ext{sqrt}{5}$ by $- ext{sqrt}{3}$:
- $- ext{sqrt}{5} imes - ext{sqrt}{3} = ext{sqrt}{5} imes ext{sqrt}{3} = ext{sqrt}{15}$ (Again, negative times negative is positive, and $ ext{sqrt}{5} imes ext{sqrt}{3}$ simplifies to $ ext{sqrt}{15}$.)
### Step 3: Combine the Results
Now we take both results from our distribution:
- From the first term, we have $10 ext{sqrt}{5}$.
- From the second term, we have $ ext{sqrt}{15}$.
When we combine these, we find that the expression simplifies to:
$$10 ext{sqrt}{5} + ext{sqrt}{15}$$
### Real-World Application
Understanding how to manipulate square roots and distribute terms is essential, not just in mathematics but in various fields such as physics, engineering, and finance. For instance, when calculating areas or volumes, you might encounter expressions involving square roots, and knowing how to simplify them will make your calculations easier and more accurate.
### Practice Example
To further enhance your understanding, try simplifying the following expression:
$$- ext{sqrt}{2}(-4 - ext{sqrt}{6})$$
Follow the same distribution steps to find the result.
### Conclusion
With practice, simplifying expressions involving square roots will become second nature. Remember, the key steps are to distribute correctly and combine like terms. If you have more questions or need further assistance, check out our resources or ask for help!
Was this answer helpful?