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. $y \sqrt{5}$
Answer (1)
Like radicals have the same index and radicand.
Examine each option to find the one with the same index (2) and radicand (5) as 3 5 .
y 5 has the same index and radicand.
The like radical is y 5 .
Explanation
Understanding Like Radicals We need to identify which of the given options is a like radical to $3
\times \sqrt{5}$. Like radicals have the same index and radicand (the number under the radical sign). In this case, the index is 2 (since it's a square root) and the radicand is 5.
Analyzing Each Option Let's examine each option:
x ( 3 5 ) : This has a cube root, so the index is 3. This is not a like radical.
5 y : This has a square root (index 2), and the radicand is 5 y . If y is a constant, this could be a like radical, but it's not exactly in the same form as 3 5 .
3 ( 3 5 x ) : This has a cube root, so the index is 3. This is not a like radical.
y 5 : This has a square root (index 2), and the radicand is 5. This is a like radical because it has the same index and radicand as 3 5 .
Identifying the Like Radical Therefore, the like radical to 3 5 is y 5 .
Examples
Like radicals are used in various mathematical contexts, such as simplifying expressions, solving equations, and performing operations with radicals. For example, if you need to add or subtract radicals, you can only combine like radicals. Imagine you're calculating the total length of two pieces of wood, one measuring 3 5 meters and the other y 5 meters. If y = 2 , the total length would be ( 3 + 2 ) 5 = 5 5 meters. Understanding like radicals simplifies such calculations.
