Distribute And Simplify The Radicals Below: { (\sqrt{12} + 6)(-\sqrt{8} - \sqrt{2})$}$

Introduction

Radicals, also known as square roots, are an essential concept in mathematics. They are used to represent the value of a number that, when multiplied by itself, gives the original number. In this article, we will focus on distributing and simplifying radicals, specifically the expression ${(\sqrt{12} + 6)(-\sqrt{8} - \sqrt{2})}$. We will break down the process into manageable steps, making it easier to understand and apply.

Understanding Radicals

Before we dive into the problem, let's quickly review what radicals are and how they work. A radical is a symbol used to represent the square root of a number. For example, ${\sqrt{16}}$ represents the number that, when multiplied by itself, gives 16. In this case, ${\sqrt{16} = 4}$ because ${4 \times 4 = 16}$.

Breaking Down the Expression

Now that we have a basic understanding of radicals, let's break down the given expression ${(\sqrt{12} + 6)(-\sqrt{8} - \sqrt{2})}$. To simplify this expression, we need to distribute the terms inside the parentheses.

Distributing the Terms

To distribute the terms, we need to multiply each term inside the first parentheses by each term inside the second parentheses. This will give us a long expression with multiple terms.

${(\sqrt{12} + 6)(-\sqrt{8} - \sqrt{2}) = -\sqrt{12} \times -\sqrt{8} - \sqrt{12} \times -\sqrt{2} - 6 \times -\sqrt{8} - 6 \times -\sqrt{2}}$

Simplifying the Expression

Now that we have distributed the terms, let's simplify the expression. We can start by simplifying the radicals.

${\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}}$

${\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}}$

Substituting these values back into the expression, we get:

${-\sqrt{12} \times -\sqrt{8} - \sqrt{12} \times -\sqrt{2} - 6 \times -\sqrt{8} - 6 \times -\sqrt{2} = -2\sqrt{3} \times -2\sqrt{2} - 2\sqrt{3} \times -\sqrt{2} - 6 \times -2\sqrt{2} - 6 \times -\sqrt{2}}$

Multiplying the Radicals

Now that we have simplified the radicals, let's multiply them together.

${-2\sqrt{3} \times -2\sqrt{2} = 4\sqrt{6}}$

${-2\sqrt{3} \times -\sqrt{2} = 2\sqrt{6}}$

${-6 \times -2\sqrt{2} = 12\sqrt{2}}$

${-6 \times -\sqrt{2} = 6\sqrt{2}}$

Combining the Terms

Now that we have multiplied the radicals, let's combine the terms.

${4\sqrt{6} + 2\sqrt{6} + 12\sqrt{2} + 6\sqrt{2} = 6\sqrt{6} + 18\sqrt{2}}$

Conclusion

In this article, we have distributed and simplified the radicals in the expression ${(\sqrt{12} + 6)(-\sqrt{8} - \sqrt{2})}$. We broke down the process into manageable steps, making it easier to understand and apply. By simplifying the radicals and multiplying them together, we arrived at the final expression ${6\sqrt{6} + 18\sqrt{2}}$.

Final Answer

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Introduction

In our previous article, we explored the process of distributing and simplifying radicals in the expression ${(\sqrt{12} + 6)(-\sqrt{8} - \sqrt{2})}$. We broke down the process into manageable steps, making it easier to understand and apply. In this article, we will answer some common questions related to distributing and simplifying radicals.

Q: What is the difference between distributing and simplifying radicals?

A: Distributing radicals involves multiplying each term inside the first parentheses by each term inside the second parentheses. Simplifying radicals, on the other hand, involves reducing the radical expression to its simplest form by factoring out perfect squares.

Q: How do I know when to distribute and when to simplify radicals?

A: When you see an expression with multiple radicals, you should first try to simplify each radical individually. If the radicals are not in their simplest form, you should simplify them before distributing. Once you have simplified the radicals, you can distribute them.

Q: What is the order of operations when distributing radicals?

A: When distributing radicals, you should follow the order of operations (PEMDAS):

1. Parentheses: Evaluate any expressions inside the parentheses.
2. Exponents: Evaluate any exponential expressions.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Evaluate any addition and subtraction operations from left to right.

Q: How do I simplify radicals with multiple terms?

A: To simplify radicals with multiple terms, you should first factor out any perfect squares from each term. Then, you can simplify the radical expression by reducing it to its simplest form.

Q: What is the difference between a perfect square and a non-perfect square?

A: A perfect square is a number that can be expressed as the product of an integer and itself. For example, 4 is a perfect square because it can be expressed as 2 x 2. A non-perfect square, on the other hand, is a number that cannot be expressed as the product of an integer and itself.

Q: How do I identify perfect squares in radical expressions?

A: To identify perfect squares in radical expressions, you should look for numbers that can be expressed as the product of an integer and itself. For example, ${\sqrt{16}}$ is a perfect square because it can be expressed as 4 x 4.

Q: What is the final answer to the expression ${(\sqrt{12} + 6)(-\sqrt{8} - \sqrt{2})}$?

A: The final answer to the expression ${(\sqrt{12} + 6)(-\sqrt{8} - \sqrt{2})}$ is ${6\sqrt{6} + 18\sqrt{2}}$.

Conclusion

In this article, we have answered some common questions related to distributing and simplifying radicals. We have covered topics such as the difference between distributing and simplifying radicals, the order of operations when distributing radicals, and how to simplify radicals with multiple terms. By following these steps and understanding the concepts, you can become more confident in your ability to distribute and simplify radicals.

Final Tips

• Always simplify radicals before distributing.
• Follow the order of operations (PEMDAS) when distributing radicals.
• Factor out perfect squares from each term to simplify radical expressions.
• Identify perfect squares in radical expressions by looking for numbers that can be expressed as the product of an integer and itself.

Final Answer

The final answer is: $\boxed{6\sqrt{6} + 18\sqrt{2}}$