Subtracting RadicalsFind The Difference: 20 − 80 \sqrt{20} - \sqrt{80} 20 ​ − 80 ​ A. − 12 5 -12 \sqrt{5} − 12 5 ​ B. − 2 5 -2 \sqrt{5} − 2 5 ​ C. 2 5 2 \sqrt{5} 2 5 ​ D. − 2 15 -2 \sqrt{15} − 2 15 ​

Introduction

Radicals, also known as square roots, are an essential part of mathematics, particularly in algebra and geometry. When dealing with radicals, it's crucial to understand how to add and subtract them correctly. In this article, we'll focus on subtracting radicals, exploring the concept, and providing a step-by-step guide on how to find the difference between two radicals.

What are Radicals?

Radicals are mathematical expressions that involve the square root of a number. They are denoted by the symbol √ and are used to represent the number that, when multiplied by itself, gives the original number. For example, √16 = 4, because 4 × 4 = 16.

Subtracting Radicals: A Conceptual Understanding

When subtracting radicals, we need to understand that the radicals must have the same index (or root) and the same radicand (the number inside the radical). If the radicals have the same index and radicand, we can subtract the numbers inside the radicals. However, if the radicals have different indices or radicands, we cannot subtract them directly.

The Concept of Like Radicals

Like radicals are radicals that have the same index and radicand. For example, √4 and √16 are like radicals because they both have the same index (square root) and the same radicand (4). On the other hand, √4 and √9 are not like radicals because they have different radicands (4 and 9).

Subtracting Like Radicals

When subtracting like radicals, we can subtract the numbers inside the radicals. For example:

$\sqrt{20} - \sqrt{80}$

To subtract these radicals, we need to find the difference between the numbers inside the radicals. We can do this by factoring the numbers inside the radicals.

Factoring the Numbers Inside the Radicals

Let's factor the numbers inside the radicals:

$\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}$

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

Now that we have factored the numbers inside the radicals, we can subtract them:

$2\sqrt{5} - 4\sqrt{5}$

Subtracting the Numbers Inside the Radicals

Now that we have the factored forms of the radicals, we can subtract the numbers inside the radicals:

$2\sqrt{5} - 4\sqrt{5} = -2\sqrt{5}$

Therefore, the difference between $\sqrt{20}$ and $\sqrt{80}$ is $-2\sqrt{5}$.

Conclusion

Subtracting radicals requires a clear understanding of the concept of like radicals and the ability to factor the numbers inside the radicals. By following the steps outlined in this article, you can confidently subtract radicals and find the difference between two radicals.

Common Mistakes to Avoid

When subtracting radicals, it's essential to avoid common mistakes such as:

• Subtracting radicals with different indices or radicands
• Not factoring the numbers inside the radicals
• Not simplifying the expression after subtracting the radicals

Practice Problems

To reinforce your understanding of subtracting radicals, try the following practice problems:

1. $\sqrt{25} - \sqrt{9}$
2. $\sqrt{36} - \sqrt{16}$
3. $\sqrt{49} - \sqrt{9}$

Answer Key

1. $4 - 3 = 1$
2. $6 - 4 = 2$
3. $7 - 3 = 4$

Final Thoughts

Introduction

In our previous article, we explored the concept of subtracting radicals and provided a step-by-step guide on how to find the difference between two radicals. However, we understand that sometimes, it's not enough to just read about a concept; you need to see it in action. That's why we've put together this Q&A guide, where we'll answer some of the most frequently asked questions about subtracting radicals.

Q: What are like radicals?

A: Like radicals are radicals that have the same index (or root) and the same radicand (the number inside the radical). For example, √4 and √16 are like radicals because they both have the same index (square root) and the same radicand (4).

Q: How do I subtract like radicals?

A: To subtract like radicals, you need to subtract the numbers inside the radicals. For example:

$\sqrt{20} - \sqrt{80}$

To subtract these radicals, you need to find the difference between the numbers inside the radicals. You can do this by factoring the numbers inside the radicals.

Q: What is the difference between $\sqrt{20}$ and $\sqrt{80}$?

A: To find the difference between $\sqrt{20}$ and $\sqrt{80}$, you need to factor the numbers inside the radicals:

$\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}$

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

Now that you have factored the numbers inside the radicals, you can subtract them:

$2\sqrt{5} - 4\sqrt{5} = -2\sqrt{5}$

Therefore, the difference between $\sqrt{20}$ and $\sqrt{80}$ is $-2\sqrt{5}$.

Q: Can I subtract radicals with different indices or radicands?

A: No, you cannot subtract radicals with different indices or radicands. For example, you cannot subtract $\sqrt{4}$ and $\sqrt{9}$ because they have different radicands (4 and 9).

Q: How do I simplify the expression after subtracting the radicals?

A: To simplify the expression after subtracting the radicals, you need to combine like terms. For example:

$2\sqrt{5} - 4\sqrt{5} = -2\sqrt{5}$

In this case, the expression is already simplified.

Q: What are some common mistakes to avoid when subtracting radicals?

A: Some common mistakes to avoid when subtracting radicals include:

• Subtracting radicals with different indices or radicands
• Not factoring the numbers inside the radicals
• Not simplifying the expression after subtracting the radicals

Q: Can you provide some practice problems to help me understand subtracting radicals?

A: Yes, here are some practice problems to help you understand subtracting radicals:

1. $\sqrt{25} - \sqrt{9}$
2. $\sqrt{36} - \sqrt{16}$
3. $\sqrt{49} - \sqrt{9}$

Answer Key

1. $4 - 3 = 1$
2. $6 - 4 = 2$
3. $7 - 3 = 4$

Conclusion

Subtracting radicals is a fundamental concept in mathematics, and it's essential to understand how to do it correctly. By following the steps outlined in this article and practicing with the provided examples, you can confidently subtract radicals and find the difference between two radicals. Remember to avoid common mistakes and always simplify the expression after subtracting the radicals.

Additional Resources

If you're looking for more practice problems or want to learn more about subtracting radicals, here are some additional resources:

• Khan Academy: Subtracting Radicals
• Mathway: Subtracting Radicals
• Purplemath: Subtracting Radicals

We hope this Q&A guide has been helpful in understanding subtracting radicals. If you have any further questions or need additional help, don't hesitate to ask.