Which Of The Following Is A Like Radical To $3 \times \sqrt{5}$?A. $x(\sqrt[3]{5}$\] B. $\sqrt{5 Y}$ C. $3(\sqrt[3]{5 X}$\] D. $y \sqrt{5}$

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What are Like Radicals?

In mathematics, like radicals are expressions that contain the same radical part. They are essential in simplifying and solving equations involving radicals. To identify like radicals, we need to look at the radical part of the expression, which is the part inside the square root or cube root symbol.

The Concept of Multiplication and Radicals

When we multiply two or more numbers, we can combine them by multiplying their coefficients (the numbers in front of the radical) and then multiplying the radicals themselves. This is a fundamental concept in algebra and is used extensively in solving equations and simplifying expressions.

The Problem at Hand

We are given the expression 3×53 \times \sqrt{5} and asked to find a like radical among the options provided. To solve this problem, we need to understand what makes two radicals like each other.

Analyzing the Options

Let's analyze each option one by one:

A. x(53x(\sqrt[3]{5}

This option contains a cube root, which is different from the square root in the given expression. Therefore, it is not a like radical.

B. 5y\sqrt{5 y}

This option contains a square root, but the coefficient is not the same as the given expression. The coefficient in the given expression is 3, while in this option, it is not explicitly mentioned. However, we can rewrite this option as y5y\sqrt{5}, which is a like radical.

C. 3(5x33(\sqrt[3]{5 x}

This option contains a cube root, which is different from the square root in the given expression. Therefore, it is not a like radical.

D. y5y \sqrt{5}

This option contains a square root, but the coefficient is not the same as the given expression. The coefficient in the given expression is 3, while in this option, it is y. However, we can rewrite this option as 5y\sqrt{5y}, which is a like radical.

Conclusion

Based on our analysis, we can conclude that the like radical to 3×53 \times \sqrt{5} is 5y\sqrt{5y} or y5y\sqrt{5}.

Why is it Important to Understand Like Radicals?

Understanding like radicals is crucial in simplifying and solving equations involving radicals. It helps us to combine radicals and simplify expressions, making it easier to solve problems. In addition, it is essential in algebra and is used extensively in solving equations and simplifying expressions.

Real-World Applications of Like Radicals

Like radicals have numerous real-world applications. They are used in physics, engineering, and other fields to simplify and solve equations involving radicals. For example, in physics, like radicals are used to simplify and solve equations involving energy and momentum.

Common Mistakes to Avoid

When working with like radicals, it is essential to avoid common mistakes. One common mistake is to confuse like radicals with unlike radicals. Unlike radicals are expressions that contain different radical parts. To avoid this mistake, we need to carefully examine the radical part of the expression and identify like radicals.

Tips and Tricks

When working with like radicals, here are some tips and tricks to keep in mind:

  • Always examine the radical part of the expression carefully.
  • Identify like radicals by looking at the radical part of the expression.
  • Combine like radicals by multiplying their coefficients and then multiplying the radicals themselves.
  • Avoid confusing like radicals with unlike radicals.

Conclusion

Q: What is the difference between like radicals and unlike radicals?

A: Like radicals are expressions that contain the same radical part, while unlike radicals are expressions that contain different radical parts.

Q: How do I identify like radicals?

A: To identify like radicals, you need to examine the radical part of the expression carefully. Look for the same radical part, such as the square root or cube root symbol, and the same radicand (the number inside the radical).

Q: Can I combine like radicals with different coefficients?

A: Yes, you can combine like radicals with different coefficients by multiplying their coefficients and then multiplying the radicals themselves.

Q: What is the rule for combining like radicals?

A: The rule for combining like radicals is to multiply their coefficients and then multiply the radicals themselves. For example, if you have two like radicals, 232\sqrt{3} and 333\sqrt{3}, you can combine them by multiplying their coefficients and then multiplying the radicals themselves: 23+33=(2+3)3=532\sqrt{3} + 3\sqrt{3} = (2+3)\sqrt{3} = 5\sqrt{3}.

Q: Can I simplify an expression with like radicals?

A: Yes, you can simplify an expression with like radicals by combining them. For example, if you have the expression 23+332\sqrt{3} + 3\sqrt{3}, you can simplify it by combining the like radicals: 23+33=532\sqrt{3} + 3\sqrt{3} = 5\sqrt{3}.

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

A: A like radical is an expression that contains the same radical part, while a similar radical is an expression that contains a similar radical part but with a different coefficient or radicand.

Q: Can I compare like radicals?

A: Yes, you can compare like radicals by comparing their coefficients and radicands. For example, if you have two like radicals, 232\sqrt{3} and 333\sqrt{3}, you can compare them by comparing their coefficients: 2<32 < 3.

Q: What is the importance of like radicals in mathematics?

A: Like radicals are essential in mathematics because they allow us to simplify and solve equations involving radicals. They are used extensively in algebra and are a fundamental concept in mathematics.

Q: Can I use like radicals in real-world applications?

A: Yes, like radicals have numerous real-world applications. They are used in physics, engineering, and other fields to simplify and solve equations involving radicals.

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 confusing like radicals with unlike radicals, not examining the radical part of the expression carefully, and not combining like radicals correctly.

Q: How can I practice working with like radicals?

A: You can practice working with like radicals by solving problems and exercises that involve combining and simplifying expressions with like radicals. You can also use online resources and math software to practice working with like radicals.

Conclusion

In conclusion, like radicals are an essential concept in mathematics that allows us to simplify and solve equations involving radicals. By understanding like radicals and how to combine and simplify them, we can become proficient in working with radicals and solve problems with ease.