To determine which expressions are like radicals to [tex]\(\sqrt{11}\)[/tex], we need to understand what like radicals are. Like radicals have the same index (the root) and the same radicand (the number under the root).
Given the task, we're looking for expressions that are similar to [tex]\(\sqrt{11}\)[/tex], specifically those that:
1. Have the same index, which is 2, since [tex]\(\sqrt{11}\)[/tex] is a square root.
2. Have the same radicand, which is 11.
Let's examine each expression:
1. [tex]\(6 \sqrt[3]{11}\)[/tex]:
- Index: 3 (cube root)
- Radicand: 11
- Conclusion: This is not a like radical because the index is different (3 vs. 2).
2. [tex]\(x \sqrt{11}\)[/tex]:
- Index: 2 (square root)
- Radicand: 11
- Conclusion: This is a like radical because both the index and the radicand match [tex]\(\sqrt{11}\)[/tex].
3. [tex]\(2 \sqrt[3]{11}\)[/tex]:
- Index: 3 (cube root)
- Radicand: 11
- Conclusion: This is not a like radical because the index is different (3 vs. 2).
4. [tex]\(-5 \sqrt[4]{11}\)[/tex]:
- Index: 4 (fourth root)
- Radicand: 11
- Conclusion: This is not a like radical because the index is different (4 vs. 2).
5. [tex]\(-6 \sqrt{11}\)[/tex]:
- Index: 2 (square root)
- Radicand: 11
- Conclusion: This is a like radical because both the index and the radicand match [tex]\(\sqrt{11}\)[/tex].
So, the expressions that are like radicals to [tex]\(\sqrt{11}\)[/tex] are:
[tex]\[
x \sqrt{11} \quad \text{and} \quad -6 \sqrt{11}
\][/tex]
Thus, the final selection of like radicals to [tex]\(\sqrt{11}\)[/tex] includes:
[tex]\[
x \sqrt{11} \quad \text{and} \quad -6 \sqrt{11}
\][/tex]
