Sure, let's simplify the given expression step by step. We need to simplify the radical expression:
[tex]\[
\sqrt[4]{162 w^{13}}
\][/tex]
First, let's break down the expression inside the radical. Notice that:
[tex]\[
162 = 2 \times 81 = 2 \times 3^4
\][/tex]
Therefore, we can rewrite the expression under the fourth root as:
[tex]\[
\sqrt[4]{2 \times 3^4 \times w^{13}}
\][/tex]
We know that the fourth root of a product can be written as the product of the fourth roots. So, we can break this down and take the fourth root of each component individually:
[tex]\[
\sqrt[4]{2 \times 3^4 \times w^{13}} = \sqrt[4]{2} \times \sqrt[4]{3^4} \times \sqrt[4]{w^{13}}
\][/tex]
Next, we simplify each part. The fourth root of [tex]\(3^4\)[/tex] is straightforward:
[tex]\[
\sqrt[4]{3^4} = 3
\][/tex]
For [tex]\(w^{13}\)[/tex], we can write:
[tex]\[
\sqrt[4]{w^{13}} = w^{13/4}
\][/tex]
Putting it all together, we have:
[tex]\[
\sqrt[4]{2} \times 3 \times w^{13/4}
\][/tex]
Therefore, the radical expression [tex]\(\sqrt[4]{162w^{13}}\)[/tex] simplifies to:
[tex]\[
3 \sqrt[4]{2} w^{13/4}
\][/tex]
Thus, the simplified radical form is:
[tex]\[
3 \cdot 2^{1/4} \cdot (w^{13})^{1/4}
\][/tex]
Which can also be written as:
[tex]\[
3 \cdot 2^{1/4} \cdot w^{13/4}
\][/tex]
And that's our final simplified form.
