Simplifying Radicals: A Step-by-Step Guide

Let's dive into simplifying the expression $2 \sqrt{2}(\sqrt{15}-\sqrt{14})$. This problem involves multiplying a radical expression by a term outside the parentheses and then simplifying the result. We'll go step by step, making sure everything is crystal clear. Our main goal here is to break down each part, apply the distributive property, and simplify any radicals that we can.

Understanding the Basics of Radical Simplification

Before we tackle the given expression, let's refresh our understanding of radicals. Radicals, often represented by the square root symbol $\sqrt{}$, indicate a root of a number. For instance, $\sqrt{9}$ equals 3 because 3 multiplied by itself equals 9. Simplifying radicals involves breaking down the number inside the square root into its prime factors and pulling out any pairs of identical factors. This process relies on understanding that $\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$, which allows us to separate and simplify.

Why is this important, guys? Simplifying radicals helps us express numbers in their simplest form, making them easier to work with in calculations and comparisons. It's a fundamental skill in algebra and is essential for more advanced math topics.

For instance, let's consider $\sqrt{8}$. We can break down 8 into $2 \cdot 2 \cdot 2$. Thus, $\sqrt{8} = \sqrt{2 \cdot 2 \cdot 2} = \sqrt{2^2 \cdot 2} = 2\sqrt{2}$. We've simplified $\sqrt{8}$ into $2\sqrt{2}$ by pulling out a pair of 2s.

Understanding prime factorization is also crucial. Prime factors are the prime numbers that divide a given number exactly. For example, the prime factors of 12 are 2 and 3 because $12 = 2 \cdot 2 \cdot 3 = 2^2 \cdot 3$. Being able to quickly identify these factors helps in simplifying radicals efficiently.

Moreover, knowing common square roots like $\sqrt{4} = 2$, $\sqrt{9} = 3$, $\sqrt{16} = 4$, $\sqrt{25} = 5$, and so on, allows for quicker simplification. These are the building blocks that make simplifying more complex radicals easier.

Step-by-Step Solution

1. Distribute $2\[sqrt]{2}$ Across the Parentheses

The first step is to distribute $2\sqrt{2}$ to both terms inside the parentheses: $(\sqrt{15} - \sqrt{14})$. This means we multiply $2\sqrt{2}$ by both $\sqrt{15}$ and $\sqrt{14}$. The expression then becomes:

$2\sqrt{2} \cdot \sqrt{15} - 2\sqrt{2} \cdot \sqrt{14}$

Why do we do this? Distributing allows us to break down a complex expression into simpler parts that we can handle individually. It’s like dividing a big task into smaller, manageable steps.

2. Multiply the Radicals

Next, we multiply the radicals together. Recall that $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$. Applying this rule, we get:

$2\sqrt{2 \cdot 15} - 2\sqrt{2 \cdot 14}$

This simplifies to:

$2\sqrt{30} - 2\sqrt{28}$

What’s happening here? We’re combining the terms under the square root to see if we can find any perfect square factors.

3. Simplify the Radicals

Now, we simplify the radicals $\sqrt{30}$ and $\sqrt{28}$ if possible. Let's start with $\sqrt{30}$. The prime factors of 30 are 2, 3, and 5. Since there are no pairs of identical factors, $\sqrt{30}$ cannot be simplified further. Thus, $\sqrt{30}$ remains as is.

Next, let's simplify $\sqrt{28}$. The prime factors of 28 are 2, 2, and 7. We can rewrite $\sqrt{28}$ as $\sqrt{2 \cdot 2 \cdot 7}$, which equals $\sqrt{2^2 \cdot 7}$. We can then pull out the pair of 2s:

$\sqrt{28} = 2\sqrt{7}$

Why simplify? Simplifying radicals helps us express the numbers in their most basic form, making them easier to understand and work with.

4. Substitute the Simplified Radicals Back into the Expression

Now that we've simplified $\sqrt{28}$ to $2\sqrt{7}$, we substitute it back into our expression:

$2\sqrt{30} - 2(2\sqrt{7})$

This simplifies to:

$2\sqrt{30} - 4\sqrt{7}$

What does this mean? We’re replacing the original term with its simplified version to make the expression cleaner.

5. Final Answer

Since $\sqrt{30}$ and $\sqrt{7}$ cannot be simplified further, and they are not like terms (i.e., they don't have the same radical), we cannot combine them. Therefore, our final simplified expression is:

$2\sqrt{30} - 4\sqrt{7}$

And that's it! We’ve simplified the given expression to its simplest form.

Common Mistakes to Avoid

1. Incorrect Distribution: Forgetting to distribute the term outside the parentheses to both terms inside. Always ensure that every term inside the parentheses is multiplied by the term outside.
2. Improper Simplification: Incorrectly factoring the numbers inside the square roots. Make sure you break down the numbers into their prime factors correctly.
3. Combining Unlike Terms: Trying to add or subtract radicals that are not like terms. Remember, you can only combine radicals if they have the same number under the square root.
4. Forgetting to Simplify Completely: Failing to simplify a radical fully. Always check if the number inside the square root has any perfect square factors that can be pulled out.
5. Arithmetic Errors: Double-check your calculations, especially when multiplying or simplifying the terms.

Practice Problems

To solidify your understanding, try simplifying these expressions:

1. $3\sqrt{3}(\sqrt{12} - \sqrt{6})$
2. $\sqrt{5}(2\sqrt{10} + \sqrt{15})$
3. $4\sqrt{2}(\sqrt{8} - \sqrt{20})$

These practice problems will give you hands-on experience and help you master the process of simplifying radical expressions.

Conclusion

Simplifying radical expressions might seem daunting at first, but by following these step-by-step instructions, you can confidently tackle these problems. Remember to distribute, multiply the radicals, simplify each radical, and combine like terms if possible. With practice, you’ll find these simplifications become second nature. So, keep practicing, and you'll become a pro at simplifying radicals in no time! You got this, guys!