To express [tex]\(\sqrt[5]{192}\)[/tex] in simplified radical form, we need to understand how to break down the number 192 into its prime factors and then simplify the expression.
1. Prime Factorization of 192:
- 192 can be factorized as follows:
[tex]\[
192 = 2 \times 96
= 2 \times 2 \times 48
= 2 \times 2 \times 2 \times 24
= 2 \times 2 \times 2 \times 2 \times 12
= 2 \times 2 \times 2 \times 2 \times 2 \times 6
= 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3
\][/tex]
- So, the prime factorization of 192 is:
[tex]\[
192 = 2^6 \times 3
\][/tex]
2. Expressing the 5th Root:
- We are looking for the 5th root of [tex]\(192\)[/tex]:
[tex]\[
\sqrt[5]{192} = \sqrt[5]{2^6 \times 3}
\][/tex]
3. Simplifying the Radical:
- The properties of roots allow us to separate the factors under the root:
[tex]\[
\sqrt[5]{2^6 \times 3} = \sqrt[5]{2^6} \times \sqrt[5]{3}
\][/tex]
- We know from exponent properties that:
[tex]\[
\sqrt[5]{2^6} = 2^{6/5} = 2 \times 2^{1/5}
\][/tex]
4. Combining the Terms:
- Now we multiply the simplified terms:
[tex]\[
2 \times \sqrt[5]{2} \times \sqrt[5]{3}
\][/tex]
- Since [tex]\(\sqrt[5]{2} \times \sqrt[5]{3} = \sqrt[5]{6}\)[/tex]:
[tex]\[
2 \times \sqrt[5]{6}
\][/tex]
Hence, the simplified radical form of [tex]\(\sqrt[5]{192}\)[/tex] is:
[tex]\[
2 \times \sqrt[5]{6}
\][/tex]
So, [tex]\(\sqrt[5]{192}\)[/tex] in simplified radical form is [tex]\(2 \cdot 6^{1/5}\)[/tex].
