Combine The Radicals { -2 \sqrt{13} + 19 \sqrt{13}$}$.A. ${ 17 \sqrt{26}\$} B. { -21 \sqrt{13}$}$C. ${ 17 \sqrt{13}\$} D. ${ 21 \sqrt{26}\$}

Understanding Radicals

Radicals are mathematical expressions that involve the square root of a number. They are commonly denoted by the symbol √. In this article, we will focus on combining radicals, which is an essential skill in mathematics.

What are Radicals?

A radical is a mathematical expression that involves the square root of a number. It is denoted by the symbol √. For example, √4 = 2, √9 = 3, and √16 = 4. Radicals can be positive or negative, and they can be combined using various operations.

Combining Radicals

Combining radicals involves adding or subtracting radicals with the same index. The index is the number that appears in front of the square root symbol. For example, √4 and √16 have the same index, which is 2.

Step 1: Identify the Radicals

To combine radicals, we need to identify the radicals that we want to combine. In this case, we have two radicals: -2√13 and 19√13.

Step 2: Check if the Radicals have the Same Index

The index of a radical is the number that appears in front of the square root symbol. In this case, both radicals have the same index, which is √13.

Step 3: Combine the Radicals

Now that we have identified the radicals and checked if they have the same index, we can combine them. To combine radicals, we add or subtract the coefficients (the numbers in front of the radicals) and keep the same index.

Combining Radicals: A Formula

The formula for combining radicals is:

a√b + c√b = (a + c)√b

where a and c are the coefficients of the radicals, and b is the index of the radical.

Applying the Formula

Using the formula, we can combine the radicals -2√13 and 19√13 as follows:

-2√13 + 19√13 = (-2 + 19)√13 = 17√13

Conclusion

Combining radicals is an essential skill in mathematics. By following the steps outlined in this article, we can combine radicals with the same index. Remember to identify the radicals, check if they have the same index, and apply the formula to combine them.

Answer

The correct answer is C. 17√13.

Why is this the Correct Answer?

This is the correct answer because we combined the radicals -2√13 and 19√13 using the formula a√b + c√b = (a + c)√b. The result is 17√13, which is the correct answer.

Tips and Tricks

• Make sure to identify the radicals and check if they have the same index before combining them.
• Use the formula a√b + c√b = (a + c)√b to combine radicals.
• Simplify the result by combining like terms.

Common Mistakes

• Combining radicals with different indices.
• Not checking if the radicals have the same index before combining them.
• Not applying the formula correctly.

Conclusion

Frequently Asked Questions

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

A: Combining radicals involves adding or subtracting radicals with the same index, while simplifying radicals involves expressing a radical in its simplest form.

Q: How do I know if two radicals have the same index?

A: To determine if two radicals have the same index, look at the number that appears in front of the square root symbol. If the numbers are the same, then the radicals have the same index.

Q: Can I combine radicals with different indices?

A: No, you cannot combine radicals with different indices. Radicals with different indices must be simplified separately.

Q: What is the formula for combining radicals?

A: The formula for combining radicals is:

a√b + c√b = (a + c)√b

where a and c are the coefficients of the radicals, and b is the index of the radical.

Q: Can I combine negative radicals?

A: Yes, you can combine negative radicals. When combining negative radicals, remember to change the sign of the result.

Q: How do I simplify a radical after combining it with another radical?

A: To simplify a radical after combining it with another radical, look for any common factors between the coefficients and the index. If you find any common factors, simplify the radical by dividing the coefficients and the index by the common factor.

Q: Can I combine radicals with variables?

A: Yes, you can combine radicals with variables. When combining radicals with variables, remember to treat the variables as coefficients.

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

A: The order of operations when combining radicals is:

1. Identify the radicals to be combined.
2. Check if the radicals have the same index.
3. Combine the radicals using the formula a√b + c√b = (a + c)√b.
4. Simplify the result by combining like terms.

Q: Can I use a calculator to combine radicals?

A: Yes, you can use a calculator to combine radicals. However, make sure to check your calculator's settings to ensure that it is set to display radicals in the correct format.

Q: How do I know if I have combined radicals correctly?

A: To check if you have combined radicals correctly, plug the result into the original equation and simplify. If the result is the same as the original equation, then you have combined the radicals correctly.

Conclusion

Combining radicals is an essential skill in mathematics. By following the steps outlined in this article, you can combine radicals with the same index. Remember to identify the radicals, check if they have the same index, and apply the formula to combine them. If you have any further questions, feel free to ask.