To determine if any of the given radicals are like radicals to [tex]\( \sqrt[3]{54} \)[/tex], we'll need to simplify each one and compare them.
1. Simplifying [tex]\( \sqrt[3]{54} \)[/tex]:
[tex]\[ \sqrt[3]{54} \][/tex]
54 does not have any perfect cubes as factors other than 1, so it remains as [tex]\( \sqrt[3]{54} \)[/tex].
2. Simplifying [tex]\( \sqrt[3]{24} \)[/tex]:
[tex]\[ \sqrt[3]{24} \][/tex]
24 does not have perfect cubes as factors other than 1, so it remains as [tex]\( \sqrt[3]{24} \)[/tex].
3. Simplifying [tex]\( \sqrt[3]{162} \)[/tex]:
[tex]\[ \sqrt[3]{162} \][/tex]
162 can be factored into [tex]\( 2 \times 81 \)[/tex], and since 81 is [tex]\( 3^4 \)[/tex], it does not reduce to a simpler form for cube roots, so it remains [tex]\( \sqrt[3]{162} \)[/tex].
4. Simplifying [tex]\( \sqrt{128} \)[/tex]:
[tex]\[ \sqrt{128} \][/tex]
128 can be factored into [tex]\( 2^7 \)[/tex]. Its square root is [tex]\( \sqrt{2^7} = 2^{7/2} \)[/tex].
5. Simplifying [tex]\( \sqrt[3]{128} \)[/tex]:
[tex]\[ \sqrt[3]{128} \][/tex]
128 can be factored into [tex]\( 2^7 \)[/tex]. Its cube root is [tex]\( \sqrt[3]{2^7} = 2^{7/3} \)[/tex].
We now compare the simplified forms:
- [tex]\( \sqrt[3]{54} \)[/tex]
- [tex]\( \sqrt[3]{24} \)[/tex]
- [tex]\( \sqrt[3]{162} \)[/tex]
- [tex]\( \sqrt{128} = 2^{7/2} \)[/tex]
- [tex]\( \sqrt[3]{128} = 2^{7/3} \)[/tex]
Given the results, none of the provided radicals simplify to give a like radical to [tex]\( \sqrt[3]{54} \)[/tex]. Therefore, the answer is:
None of the provided expressions are like radicals to [tex]\( \sqrt[3]{54} \)[/tex].
