Which of the following is a like radical to [tex]$3 x \sqrt{5}$[/tex]?
[tex]$\sqrt{5 y}$[/tex]
[tex]$3(\sqrt[3]{5 x})$[/tex]
[tex]$x(\sqrt[3]{5})$[/tex]
[tex]$y \sqrt{5}$[/tex]
Answer (2)
Like radicals have the same index and radicand.
The given radical 3 x 5 has an index of 2 and a radicand of 5.
Examine each option to find one with the same index and radicand.
The like radical is y 5 .
Explanation
Understanding Like Radicals We need to identify which of the given radicals is 'like' the radical 3 x 5 . Like radicals have the same index and radicand (the number under the radical). Let's break down the given radical:
Identifying Index and Radicand The given radical is 3 x 5 . The index is 2 (since it's a square root), and the radicand is 5. We need to find an option with the same index and radicand.
Checking Each Option Let's examine each option:
5 y : The index is 2, and the radicand is 5 y . This is NOT a like radical because the radicand is different.
3 ( 3 5 x ) : The index is 3, and the radicand is 5 x . This is NOT a like radical because the index is different.
x ( 3 5 ) : The index is 3, and the radicand is 5. This is NOT a like radical because the index is different.
y 5 : The index is 2, and the radicand is 5. This IS a like radical because both the index and radicand are the same.
Conclusion Therefore, the like radical to 3 x 5 is y 5 .
Examples
Understanding like radicals is essential in simplifying expressions, just like grouping similar objects in real life. For instance, if you're counting apples and oranges, you can only add apples to apples and oranges to oranges. Similarly, in algebra, you can only combine like radicals. Imagine you are calculating the total length of wooden planks: if you have 3 x 5 meters of one type of plank and y 5 meters of the same type, you can easily add them together because they are 'like' terms. This concept is crucial in construction, engineering, and any field where precise measurements and simplifications are necessary.
The like radical to 3 x 5 is y 5 because it shares the same index of 2 and the same radicand of 5. The other options either have a different index or a different radicand. Therefore, the correct answer is y 5 .
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