To simplify the expression [tex]\(\sqrt[4]{162 u^5}\)[/tex], follow these steps:
1. Factorize 162:
First, we express 162 as the product of its prime factors:
[tex]\[
162 = 2 \times 81 = 2 \times 3^4
\][/tex]
Hence, we can write:
[tex]\[
162 = 2 \times 3^4
\][/tex]
2. Rewrite the expression:
Substitute the factorization into the original expression:
[tex]\[
\sqrt[4]{162 u^5} = \sqrt[4]{2 \times 3^4 \times u^5}
\][/tex]
3. Separate the terms under the fourth root:
Using properties of radicals, we can separate the terms:
[tex]\[
\sqrt[4]{2 \times 3^4 \times u^5} = \sqrt[4]{2} \times \sqrt[4]{3^4} \times \sqrt[4]{u^5}
\][/tex]
4. Simplify each part:
a. Simplify [tex]\(\sqrt[4]{3^4}\)[/tex]:
[tex]\[
\sqrt[4]{3^4} = 3
\][/tex]
b. Simplify [tex]\(\sqrt[4]{u^5}\)[/tex]:
[tex]\[
\sqrt[4]{u^5} = (u^5)^{1/4} = u^{5/4}
\][/tex]
5. Combine the simplified terms:
Multiply the simplified parts together:
[tex]\[
\sqrt[4]{2} \times 3 \times u^{5/4}
\][/tex]
Therefore, the simplified radical form of the expression is:
[tex]\[
\sqrt[4]{162 u^5} = 3 \sqrt[4]{2} u^{5/4}
\][/tex]
