Certainly! Let's simplify the given cube roots and identify their radical parts.
1. Simplifying [tex]\(\sqrt[3]{54}\)[/tex]:
To simplify [tex]\(\sqrt[3]{54}\)[/tex]:
- Begin by finding the prime factorization of 54:
[tex]\[
54 = 2 \times 3 \times 3 \times 3 = 2 \times 3^3
\][/tex]
- Rewrite the expression using these factors:
[tex]\[
\sqrt[3]{54} = \sqrt[3]{2 \times 3^3}
\][/tex]
- We can separate the cube root as follows:
[tex]\[
\sqrt[3]{2 \times 3^3} = \sqrt[3]{2} \times \sqrt[3]{3^3}
\][/tex]
- Since [tex]\(\sqrt[3]{3^3} = 3\)[/tex], simplify:
[tex]\[
\sqrt[3]{54} = 3 \times \sqrt[3]{2}
\][/tex]
Hence, the radical part of [tex]\(\sqrt[3]{54}\)[/tex] is:
[tex]\[
\sqrt[3]{2}
\][/tex]
2. Simplifying [tex]\(\sqrt[3]{128}\)[/tex]:
To simplify [tex]\(\sqrt[3]{128}\)[/tex]:
- Begin by finding the prime factorization of 128:
[tex]\[
128 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^7
\][/tex]
- Rewrite the expression using these factors:
[tex]\[
\sqrt[3]{128} = \sqrt[3]{2^7}
\][/tex]
- We can separate the cube root as follows:
[tex]\[
\sqrt[3]{2^7} = \sqrt[3]{2^6 \times 2} = \sqrt[3]{(2^3)^2 \times 2}
\][/tex]
- Since [tex]\(\sqrt[3]{(2^3)^2} = 2^2 = 4\)[/tex], simplify:
[tex]\[
\sqrt[3]{2^7} = 4 \times \sqrt[3]{2}
\][/tex]
Hence, the radical part of [tex]\(\sqrt[3]{128}\)[/tex] is:
[tex]\[
\sqrt[3]{2}
\][/tex]
Thus, for both [tex]\(\sqrt[3]{54}\)[/tex] and [tex]\(\sqrt[3]{128}\)[/tex], the radical part when the expressions are simplified is:
[tex]\[
\sqrt[3]{2}
\][/tex]
