To determine which of the given expressions are like radicals to [tex]\(\sqrt{11}\)[/tex], we need to compare the expressions' radical parts with [tex]\(\sqrt{11}\)[/tex], or [tex]\(11^{1/2}\)[/tex]. Like radicals have the same radicand (the number under the radical) and the same index (the root).
Given the expressions:
- [tex]\(6 \sqrt[3]{11}\)[/tex]
- [tex]\(x \sqrt{11}\)[/tex]
- [tex]\(2 \sqrt[3]{11}\)[/tex]
- [tex]\(-5 \sqrt[4]{11}\)[/tex]
- [tex]\(-6 \sqrt{11}\)[/tex]
Let's break them down:
1. [tex]\(6 \sqrt[3]{11}\)[/tex]:
- This is [tex]\(6 \times 11^{1/3}\)[/tex].
- The radicand is [tex]\(11\)[/tex], and the index is [tex]\(3\)[/tex].
2. [tex]\(x \sqrt{11}\)[/tex]:
- This is [tex]\(x \times 11^{1/2}\)[/tex].
- The radicand is [tex]\(11\)[/tex], and the index is [tex]\(2\)[/tex].
3. [tex]\(2 \sqrt[3]{11}\)[/tex]:
- This is [tex]\(2 \times 11^{1/3}\)[/tex].
- The radicand is [tex]\(11\)[/tex], and the index is [tex]\(3\)[/tex].
4. [tex]\(-5 \sqrt[4]{11}\)[/tex]:
- This is [tex]\(-5 \times 11^{1/4}\)[/tex].
- The radicand is [tex]\(11\)[/tex], and the index is [tex]\(4\)[/tex].
5. [tex]\(-6 \sqrt{11}\)[/tex]:
- This is [tex]\(-6 \times 11^{1/2}\)[/tex].
- The radicand is [tex]\(11\)[/tex], and the index is [tex]\(2\)[/tex].
To be like radicals to [tex]\(\sqrt{11}\)[/tex] (which is [tex]\(11^{1/2}\)[/tex]), the expressions must have the same radicand [tex]\(11\)[/tex] and same index [tex]\(2\)[/tex].
Comparing all the given expressions, the ones that match [tex]\(\sqrt{11}\)[/tex] are:
- [tex]\(x \sqrt{11}\)[/tex]
- [tex]\(-6 \sqrt{11}\)[/tex]
So, the expressions like radicals to [tex]\(\sqrt{11}\)[/tex] are:
[tex]\[ x \sqrt{11} \][/tex]
[tex]\[ -6 \sqrt{11} \][/tex]
