Simplify The Expression:$\[ -3 \sqrt{24} + 2 \sqrt{5} - 2 \sqrt{54} - 2 \sqrt{80} \\]

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Introduction

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Simplifying radical expressions is a crucial skill in mathematics, particularly in algebra and geometry. It involves combining like terms and simplifying the expression to its simplest form. In this article, we will focus on simplifying the given expression: $-3 \sqrt{24} + 2 \sqrt{5} - 2 \sqrt{54} - 2 \sqrt{80}$. We will break down the process into manageable steps and provide a clear explanation of each step.

Step 1: Simplify the Square Roots

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The first step in simplifying the expression is to simplify the square roots. We can start by simplifying $\sqrt{24}$, $\sqrt{54}$, and $\sqrt{80}$.

Simplifying $\sqrt{24}$

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$\sqrt{24}$ can be simplified by finding the largest perfect square that divides 24. We can write 24 as $4 \times 6$. Since 4 is a perfect square, we can simplify $\sqrt{24}$ as follows:

$\sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6} = 2\sqrt{6}$

Simplifying $\sqrt{54}$

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$\sqrt{54}$ can be simplified by finding the largest perfect square that divides 54. We can write 54 as $9 \times 6$. Since 9 is a perfect square, we can simplify $\sqrt{54}$ as follows:

$\sqrt{54} = \sqrt{9 \times 6} = \sqrt{9} \times \sqrt{6} = 3\sqrt{6}$

Simplifying $\sqrt{80}$

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$\sqrt{80}$ can be simplified by finding the largest perfect square that divides 80. We can write 80 as $16 \times 5$. Since 16 is a perfect square, we can simplify $\sqrt{80}$ as follows:

$\sqrt{80} = \sqrt{16 \times 5} = \sqrt{16} \times \sqrt{5} = 4\sqrt{5}$

Step 2: Substitute the Simplified Square Roots

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Now that we have simplified the square roots, we can substitute them back into the original expression.

$-3 \sqrt{24} + 2 \sqrt{5} - 2 \sqrt{54} - 2 \sqrt{80} = -3(2\sqrt{6}) + 2\sqrt{5} - 2(3\sqrt{6}) - 2(4\sqrt{5})$

Step 3: Distribute the Coefficients

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The next step is to distribute the coefficients to the simplified square roots.

$-3(2\sqrt{6}) + 2\sqrt{5} - 2(3\sqrt{6}) - 2(4\sqrt{5}) = -6\sqrt{6} + 2\sqrt{5} - 6\sqrt{6} - 8\sqrt{5}$

Step 4: Combine Like Terms

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The final step is to combine like terms. We can combine the like terms $-6\sqrt{6}$ and $-6\sqrt{6}$, and $2\sqrt{5}$ and $-8\sqrt{5}$.

$-6\sqrt{6} + 2\sqrt{5} - 6\sqrt{6} - 8\sqrt{5} = -12\sqrt{6} - 6\sqrt{5}$

Conclusion

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In conclusion, we have simplified the given expression: $-3 \sqrt{24} + 2 \sqrt{5} - 2 \sqrt{54} - 2 \sqrt{80}$. We broke down the process into manageable steps and provided a clear explanation of each step. By simplifying the square roots, substituting them back into the original expression, distributing the coefficients, and combining like terms, we arrived at the simplified expression: $-12\sqrt{6} - 6\sqrt{5}$.

Final Answer

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The final answer is: $\boxed{-12\sqrt{6} - 6\sqrt{5}}$

Frequently Asked Questions

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Q: What is the process of simplifying radical expressions?

A: The process of simplifying radical expressions involves combining like terms and simplifying the expression to its simplest form.

Q: How do I simplify a square root?

A: To simplify a square root, find the largest perfect square that divides the number inside the square root.

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 with itself, while a non-perfect square is a number that cannot be expressed as the product of an integer with itself.

Q: How do I combine like terms?

A: To combine like terms, add or subtract the coefficients of the like terms.

References

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• [1] "Simplifying Radical Expressions" by Math Open Reference
• [2] "Simplifying Square Roots" by Khan Academy
• [3] "Combining Like Terms" by Purplemath
  

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Introduction

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Simplifying radical expressions is a crucial skill in mathematics, particularly in algebra and geometry. In our previous article, we provided a step-by-step guide to simplifying the expression: $-3 \sqrt{24} + 2 \sqrt{5} - 2 \sqrt{54} - 2 \sqrt{80}$. In this article, we will provide a Q&A guide to help you understand the process of simplifying radical expressions.

Q&A

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Q: What is the process of simplifying radical expressions?

A: The process of simplifying radical expressions involves combining like terms and simplifying the expression to its simplest form.

Q: How do I simplify a square root?

A: To simplify a square root, find the largest perfect square that divides the number inside the square root.

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 with itself, while a non-perfect square is a number that cannot be expressed as the product of an integer with itself.

Q: How do I combine like terms?

A: To combine like terms, add or subtract the coefficients of the like terms.

Q: What are some common mistakes to avoid when simplifying radical expressions?

A: Some common mistakes to avoid when simplifying radical expressions include:

• Not simplifying the square roots
• Not combining like terms
• Not checking for perfect squares
• Not using the correct order of operations

Q: How do I check if a number is a perfect square?

A: To check if a number is a perfect square, find the square root of the number and check if it is an integer.

Q: What is the difference between a rational number and an irrational number?

A: A rational number is a number that can be expressed as the ratio of two integers, while an irrational number is a number that cannot be expressed as the ratio of two integers.

Q: How do I simplify a radical expression with multiple terms?

A: To simplify a radical expression with multiple terms, simplify each term separately and then combine the like terms.

Q: What are some real-world applications of simplifying radical expressions?

A: Some real-world applications of simplifying radical expressions include:

• Calculating distances and lengths in geometry and trigonometry
• Solving problems in physics and engineering
• Working with financial data and statistics

Tips and Tricks

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Tip 1: Simplify the square roots first

Simplifying the square roots first can make it easier to combine like terms and simplify the expression.

Tip 2: Use the correct order of operations

Using the correct order of operations (PEMDAS) can help you avoid mistakes and simplify the expression correctly.

Tip 3: Check for perfect squares

Checking for perfect squares can help you simplify the expression and avoid mistakes.

Conclusion

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In conclusion, simplifying radical expressions is a crucial skill in mathematics, particularly in algebra and geometry. By following the steps outlined in this article and avoiding common mistakes, you can simplify radical expressions with ease. Remember to simplify the square roots first, use the correct order of operations, and check for perfect squares.

Final Answer

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The final answer is: $\boxed{-12\sqrt{6} - 6\sqrt{5}}$

Frequently Asked Questions

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Q: What is the process of simplifying radical expressions?

A: The process of simplifying radical expressions involves combining like terms and simplifying the expression to its simplest form.

Q: How do I simplify a square root?

A: To simplify a square root, find the largest perfect square that divides the number inside the square root.

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 with itself, while a non-perfect square is a number that cannot be expressed as the product of an integer with itself.

Q: How do I combine like terms?

A: To combine like terms, add or subtract the coefficients of the like terms.

References

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• [1] "Simplifying Radical Expressions" by Math Open Reference
• [2] "Simplifying Square Roots" by Khan Academy
• [3] "Combining Like Terms" by Purplemath