Question

three radical expressions have different radicands and, when simplified, are like radicals to \\(\sqrt{3xy}\\). which statements describe the key characteristics of these radical expressions? choose three correct answers.

the numeric coefficient of the radicand is three times a perfect-square number.

the index is 2.

the variable \\(x\\) in the radicand is raised to an odd power.

the index is 3.

the variable \\(y\\) in the radicand is raised to an even power.

Explanation:

1. Analyze "The numeric coefficient of the radicand is three times a perfect - square number":

• For a radical to simplify to a like radical of \(\sqrt{3xy}\), the numeric part of the radicand (before simplification) should be such that when simplified, the non - square part is 3. If the numeric coefficient of the radicand is \(3\times k^{2}\) (where \(k^{2}\) is a perfect - square number), then \(\sqrt{3k^{2}xy}=k\sqrt{3xy}\), which is a like radical to \(\sqrt{3xy}\). So this statement is correct.

1. Analyze "The index is 2":

• The radical \(\sqrt{3xy}\) has an index of 2 (since it's a square root). For two radicals to be like radicals, they must have the same index. So if the other radical expressions simplify to a like radical of \(\sqrt{3xy}\), their index must also be 2. This statement is correct.

1. Analyze "The variable \(x\) in the radicand is raised to an odd power":

• When simplifying a square root (\(n = 2\)), for a variable \(x\) in the radicand, if the exponent of \(x\) is odd, say \(x^{2m + 1}\) (where \(m\) is a non - negative integer), then \(\sqrt{x^{2m+1}}=x^{m}\sqrt{x}\). In the case of \(\sqrt{3xy}\), if the original radicand has \(x\) with an odd power, after simplification, the remaining \(x\) under the square root will be part of the like radical structure. So this statement is correct.

1. Analyze "The index is 3":

• The radical \(\sqrt{3xy}\) has an index of 2. A radical with an index of 3 (a cube root) cannot be a like radical to a square root. So this statement is incorrect.

1. Analyze "The variable \(y\) in the radicand is raised to an even power":

• If \(y\) is raised to an even power, say \(y^{2m}\) (where \(m\) is a positive integer), then \(\sqrt{y^{2m}}=y^{m}\), and the simplified radical would not have a \(y\) under the square root in the same way as \(\sqrt{3xy}\) (where \(y\) has an exponent of 1, an odd power). So this statement is incorrect.

Answer:

• The numeric coefficient of the radicand is three times a perfect - square number.
• The index is 2.
• The variable \(x\) in the radicand is raised to an odd power.