Verified Answer :

  • Like radicals must have the same index and the same radicand.
  • For the expression \(3x\sqrt{5}\), the index is 2 and the radicand is 5.
  • Comparing the options, \(y\sqrt{5}\) is the only expression with index 2 and radicand 5.
  • Final Answer: \(\boxed{y\sqrt{5}}\).

Explanation

  1. Understanding the Concept of Like Radicals

The objective is to identify which of the given expressions is a "like radical" to the expression \(3x\sqrt{5}\). First, we must understand the definition of like radicals. Two radicals are considered "like" if they have the same index (the root order) and the same radicand (the value inside the radical). The numerical or variable coefficient in front of the radical does not affect whether radicals are "like". For the given expression \(3x\sqrt{5}\), the index is 2 (square root) and the radicand is 5.

  1. Evaluating Each Option

Now we analyze each of the provided options to determine their index and radicand:

  1. \(x\sqrt[3]{5}\): The index is 3 (cube root) and the radicand is 5.
  2. \(\sqrt{5y}\): The index is 2 (square root) and the radicand is \(5y\).
  3. \(3\sqrt[3]{5x}\): The index is 3 (cube root) and the radicand is \(5x\).
  4. \(y\sqrt{5}\): The index is 2 (square root) and the radicand is 5.
  1. Identifying the Like Radical

Comparing each option to our base expression \(3x\sqrt{5}\), which has an index of 2 and a radicand of 5:

  • \(x\sqrt[3]{5}\) has a different index.
  • \(\sqrt{5y}\) has a different radicand.
  • \(3\sqrt[3]{5x}\) has a different index and a different radicand.
  • \(y\sqrt{5}\) has an index of 2 and a radicand of 5.

Since \(y\sqrt{5}\) shares both the same index and the same radicand with \(3x\sqrt{5}\), it is the correct "like radical".

Examples

In mathematics, like radicals are essential for simplifying expressions, just as "like terms" are in algebra. For example, when you have dimensions of a rectangle like \(y\sqrt{5}\) and \(3x\sqrt{5}\), calculating the perimeter involves adding these terms: \(P = 2(y\sqrt{5} + 3x\sqrt{5}) = 2(y + 3x)\sqrt{5}\). Knowing how to identify like radicals allows you to combine measurements and simplify complex expressions efficiently in geometry and physics.

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