Properties of Like Radicals

Oct 13, 2025 by miodeya.com 134 views

Introduction to Like Radicals

In mathematics, like radicals are expressions that contain the same type of radical, such as a square root or a cube root, and have the same radicand. The radicand is the number or expression inside the radical symbol. When we have like radicals, we can add or subtract them by combining their coefficients. In this article, we will explore which of the given options is a like radical to $3 \times \sqrt{5}$ .

What are Like Radicals?

Like radicals are expressions that have the same type of radical and the same radicand. For example, $\sqrt{5}$ and $\sqrt{5}$ are like radicals because they have the same type of radical (square root) and the same radicand (5). Similarly, $\sqrt[3]{5}$ and $\sqrt[3]{5}$ are like radicals because they have the same type of radical (cube root) and the same radicand (5).

Properties of Like Radicals

When we have like radicals, we can add or subtract them by combining their coefficients. For example, $\sqrt{5} + \sqrt{5} = 2\sqrt{5}$ and $\sqrt[3]{5} - \sqrt[3]{5} = 0$ . This means that we can combine the coefficients of like radicals to simplify the expression.

Analyzing the Options

Now, let's analyze the options given to determine which one is a like radical to $3 \times \sqrt{5}$ .

Option A: $x(\sqrt[3]{5})$

This option is not a like radical to $3 \times \sqrt{5}$ because it has a cube root instead of a square root. Additionally, the radicand is 5, but the coefficient is x, which is not the same as the coefficient in the original expression.

Option B: $\sqrt{5y}$

This option is not a like radical to $3 \times \sqrt{5}$ because it has a different radicand (5y) and a different coefficient (y). The original expression has a coefficient of 3 and a radicand of 5, so this option does not match.

Option C: $3(\sqrt[3]{5x})$

This option is not a like radical to $3 \times \sqrt{5}$ because it has a cube root instead of a square root. Additionally, the radicand is 5x, which is not the same as the radicand in the original expression.

Option D: $y\sqrt{5}$

This option is a like radical to $3 \times \sqrt{5}$ because it has the same type of radical (square root) and the same radicand (5). The only difference is the coefficient, which is y instead of 3. However, this option is still a like radical because it has the same type of radical and the same radicand.

Conclusion

In conclusion, the correct answer is Option D: $y\sqrt{5}$ . This option is a like radical to $3 \times \sqrt{5}$ because it has the same type of radical (square root) and the same radicand (5). The only difference is the coefficient, which is y instead of 3.

Final Thoughts

Understanding like radicals is an important concept in mathematics, and it can be applied to a wide range of problems. By recognizing like radicals, we can simplify expressions and solve equations more easily. In this article, we have explored the concept of like radicals and analyzed the options to determine which one is a like radical to $3 \times \sqrt{5}$ . We hope that this article has provided a clear understanding of like radicals and has helped to clarify any confusion.

Common Mistakes to Avoid

When working with like radicals, it's easy to make mistakes. Here are some common mistakes to avoid:

Not recognizing like radicals: Make sure to recognize like radicals by checking the type of radical and the radicand.
Not combining coefficients: When combining like radicals, make sure to combine the coefficients as well.
Not simplifying expressions: Simplify expressions by combining like radicals and combining coefficients.

Real-World Applications

Like radicals have many real-world applications. Here are a few examples:

Mathematics: Like radicals are used in mathematics to simplify expressions and solve equations.
Science: Like radicals are used in science to calculate quantities and solve problems.
Engineering: Like radicals are used in engineering to design and build structures.

Conclusion

In conclusion, like radicals are an important concept in mathematics, and they have many real-world applications. By understanding like radicals, we can simplify expressions and solve equations more easily. We hope that this article has provided a clear understanding of like radicals and has helped to clarify any confusion.

Q: What is a like radical?

A: A like radical is an expression that contains the same type of radical, such as a square root or a cube root, and has the same radicand. For example, $\sqrt{5}$ and $\sqrt{5}$ are like radicals because they have the same type of radical (square root) and the same radicand (5).

Q: How do I recognize like radicals?

A: To recognize like radicals, check the type of radical and the radicand. If the type of radical and the radicand are the same, then the expressions are like radicals.

Q: Can I add or subtract like radicals?

A: Yes, you can add or subtract like radicals by combining their coefficients. For example, $\sqrt{5} + \sqrt{5} = 2\sqrt{5}$ and $\sqrt[3]{5} - \sqrt[3]{5} = 0$ .

Q: Can I multiply or divide like radicals?

A: Yes, you can multiply or divide like radicals by multiplying or dividing their coefficients. For example, $2\sqrt{5} \times 3\sqrt{5} = 6\sqrt{25}$ and $\frac{\sqrt{5}}{2} \times \frac{\sqrt{5}}{2} = \frac{5}{4}$ .

Q: What is the difference between a like radical and a unlike radical?

A: A like radical is an expression that contains the same type of radical and has the same radicand, while a unlike radical is an expression that contains a different type of radical or has a different radicand.

Q: Can I simplify expressions with unlike radicals?

A: No, you cannot simplify expressions with unlike radicals. Unlike radicals must be left as is, and you cannot combine their coefficients.

Q: How do I simplify expressions with like radicals?

A: To simplify expressions with like radicals, combine their coefficients and simplify the radicand if possible.

Q: Can I use like radicals to solve equations?

A: Yes, you can use like radicals to solve equations. By combining like radicals, you can simplify the equation and solve for the variable.

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 recognizing like radicals
Not combining coefficients
Not simplifying expressions
Using unlike radicals to simplify expressions

Q: How do I apply like radicals in real-world situations?

A: Like radicals are used in many real-world situations, including:

Mathematics: Like radicals are used to simplify expressions and solve equations.
Science: Like radicals are used to calculate quantities and solve problems.
Engineering: Like radicals are used to design and build structures.

Q: Can I use like radicals to solve problems in other subjects?

A: Yes, you can use like radicals to solve problems in other subjects, including science and engineering. By applying the concept of like radicals, you can simplify expressions and solve problems more easily.

Q: What are some examples of like radicals in real-world situations?

A: Some examples of like radicals in real-world situations include:

Calculating the area of a square root of a number
Finding the volume of a cube root of a number
Simplifying expressions with like radicals in science and engineering problems

Q: Can I use like radicals to solve problems with variables?

A: Yes, you can use like radicals to solve problems with variables. By applying the concept of like radicals, you can simplify expressions and solve problems more easily.

Q: What are some tips for working with like radicals?

A: Some tips for working with like radicals include:

Recognizing like radicals by checking the type of radical and the radicand
Combining coefficients when adding or subtracting like radicals
Simplifying expressions by combining like radicals and simplifying the radicand if possible
Applying like radicals to solve problems in other subjects, including science and engineering.