Let's start by understanding what it means for terms to be like radicals to [tex]\(\sqrt{11}\)[/tex]. Terms are considered like radicals if they have the same radical part. In this context, we are looking for terms involving [tex]\(\sqrt{11}\)[/tex], which can also be written as [tex]\(11^{1/2}\)[/tex].
The options given are:
1. [tex]\(6 \sqrt[3]{11}\)[/tex]
2. [tex]\(x \sqrt{11}\)[/tex]
3. [tex]\(2 \sqrt[3]{11}\)[/tex]
4. [tex]\(-5 \sqrt[4]{11}\)[/tex]
5. [tex]\(-6 \sqrt{11}\)[/tex]
Let's examine each option to determine if the term has the same radical part as [tex]\(\sqrt{11}\)[/tex]:
1. [tex]\(6 \sqrt[3]{11}\)[/tex]: This is [tex]\(6 \times 11^{1/3}\)[/tex], which is the cubic root of 11. It does not match [tex]\(\sqrt{11}\)[/tex].
2. [tex]\(x \sqrt{11}\)[/tex]: This is [tex]\(x \times 11^{1/2}\)[/tex], which is the same radical part as [tex]\(\sqrt{11}\)[/tex]. So, this is a like radical to [tex]\(\sqrt{11}\)[/tex].
3. [tex]\(2 \sqrt[3]{11}\)[/tex]: This is [tex]\(2 \times 11^{1/3}\)[/tex], which is the cubic root of 11. It does not match [tex]\(\sqrt{11}\)[/tex].
4. [tex]\(-5 \sqrt[4]{11}\)[/tex]: This is [tex]\(-5 \times 11^{1/4}\)[/tex], which is the fourth root of 11. It does not match [tex]\(\sqrt{11}\)[/tex].
5. [tex]\(-6 \sqrt{11}\)[/tex]: This is [tex]\(-6 \times 11^{1/2}\)[/tex], which matches the same radical part as [tex]\(\sqrt{11}\)[/tex]. So, this is also a like radical to [tex]\(\sqrt{11}\)[/tex].
After examining each option, the terms that are like radicals to [tex]\(\sqrt{11}\)[/tex] are:
[tex]\[ x \sqrt{11} \quad \text{and} \quad -6 \sqrt{11} \][/tex]
Thus, the indices of the options that are like radicals to [tex]\(\sqrt{11}\)[/tex] are [tex]\(2\)[/tex] and [tex]\(5\)[/tex].
So, the correct answer is:
[tex]\[ [2, 5] \][/tex]
