Which Of The Following Are Like Radicals? Check All Of The Boxes That Apply.A. $3x \sqrt{x^2 Y}$ B. $-12x \sqrt{x^2 Y}$ C. $-2x \sqrt{xy^2}$ D. $x \sqrt{yx^2}$ E. $-x \sqrt{x^2 Y^2}$ F. $2 \sqrt{x^2

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What are Like Radicals?

In mathematics, like radicals are expressions that contain the same type of radical, which is a root of a number. The key characteristic of like radicals is that they have the same index (or root) and the same radicand (the number inside the radical). When we have like radicals, we can combine them by adding or subtracting their coefficients.

Checking for Like Radicals

To determine if two or more radicals are like radicals, we need to compare their indices and radicands. If they have the same index and radicand, then they are like radicals.

Analyzing the Options

Let's analyze each of the given options to determine if they are like radicals.

A. 3xx2y3x \sqrt{x^2 y}

This expression contains a radical with an index of 2 and a radicand of x2yx^2 y. The coefficient is 3x3x. To check if this is a like radical, we need to compare it with other options.

B. −12xx2y-12x \sqrt{x^2 y}

This expression also contains a radical with an index of 2 and a radicand of x2yx^2 y. The coefficient is −12x-12x. Since the index and radicand are the same as in option A, this is a like radical.

C. −2xxy2-2x \sqrt{xy^2}

This expression contains a radical with an index of 2 and a radicand of xy2xy^2. The coefficient is −2x-2x. Although the index is the same as in options A and B, the radicand is different, so this is not a like radical.

D. xyx2x \sqrt{yx^2}

This expression contains a radical with an index of 2 and a radicand of yx2yx^2. The coefficient is xx. Although the index is the same as in options A and B, the radicand is different, so this is not a like radical.

E. −xx2y2-x \sqrt{x^2 y^2}

This expression contains a radical with an index of 2 and a radicand of x2y2x^2 y^2. The coefficient is −x-x. Since the index and radicand are the same as in options A and B, this is a like radical.

F. 2x22 \sqrt{x^2}

This expression contains a radical with an index of 2 and a radicand of x2x^2. The coefficient is 22. Although the index is the same as in options A, B, and E, the radicand is different, so this is not a like radical.

Conclusion

Based on our analysis, the like radicals are:

  • Option A: 3xx2y3x \sqrt{x^2 y}
  • Option B: −12xx2y-12x \sqrt{x^2 y}
  • Option E: −xx2y2-x \sqrt{x^2 y^2}

Q: What is the main characteristic of like radicals?

A: The main characteristic of like radicals is that they have the same index (or root) and the same radicand (the number inside the radical).

Q: How do I determine if two or more radicals are like radicals?

A: To determine if two or more radicals are like radicals, you need to compare their indices and radicands. If they have the same index and radicand, then they are like radicals.

Q: Can I add or subtract like radicals?

A: Yes, you can add or subtract like radicals. When you add or subtract like radicals, you combine their coefficients.

Q: What is an example of adding like radicals?

A: Here's an example of adding like radicals:

16+16=4+4=8\sqrt{16} + \sqrt{16} = 4 + 4 = 8

In this example, both radicals have the same index (2) and radicand (16), making them like radicals. When we add their coefficients, we get 8.

Q: What is an example of subtracting like radicals?

A: Here's an example of subtracting like radicals:

25−25=5−5=0\sqrt{25} - \sqrt{25} = 5 - 5 = 0

In this example, both radicals have the same index (2) and radicand (25), making them like radicals. When we subtract their coefficients, we get 0.

Q: Can I multiply or divide like radicals?

A: Yes, you can multiply or divide like radicals. When you multiply or divide like radicals, you multiply or divide their coefficients.

Q: What is an example of multiplying like radicals?

A: Here's an example of multiplying like radicals:

16×16=4×4=16\sqrt{16} \times \sqrt{16} = 4 \times 4 = 16

In this example, both radicals have the same index (2) and radicand (16), making them like radicals. When we multiply their coefficients, we get 16.

Q: What is an example of dividing like radicals?

A: Here's an example of dividing like radicals:

36÷36=6÷6=1\sqrt{36} \div \sqrt{36} = 6 \div 6 = 1

In this example, both radicals have the same index (2) and radicand (36), making them like radicals. When we divide their coefficients, we get 1.

Q: What are some common mistakes to avoid when working with like radicals?

A: Some common mistakes to avoid when working with like radicals include:

  • Not comparing the indices and radicands of the radicals
  • Not combining the coefficients of like radicals
  • Not multiplying or dividing the coefficients of like radicals

Q: How can I practice working with like radicals?

A: You can practice working with like radicals by:

  • Writing examples of like radicals and combining their coefficients
  • Creating your own examples of like radicals and solving them
  • Using online resources or math textbooks to practice working with like radicals

Conclusion

Working with like radicals can be a challenging but rewarding topic in mathematics. By understanding the characteristics of like radicals and practicing working with them, you can become more confident and proficient in solving math problems involving radicals.