To solve the expression [tex]\(\sqrt[4]{3^4 b^1}\)[/tex], we will break down the problem step-by-step.
1. Understand the Problem:
We need to evaluate the fourth root of [tex]\(3^4 \cdot b^1\)[/tex].
2. Simplify the Expression:
- Compute [tex]\(3^4\)[/tex]:
[tex]\[
3^4 = 81
\][/tex]
- Now, the expression becomes:
[tex]\[
\sqrt[4]{81 \cdot b}
\][/tex]
3. Combine the Terms Under the Root:
- Keep [tex]\(b\)[/tex] as it is and note that [tex]\(81 = 3^4\)[/tex].
4. Rewrite the Radicand:
- Substitute [tex]\(81\)[/tex] with [tex]\(3^4\)[/tex]:
[tex]\[
\sqrt[4]{3^4 \cdot b}
\][/tex]
- Then you realize that [tex]\(\sqrt[4]{3^4 \cdot b}\)[/tex] simplifies because the fourth root and the exponent of 4 cancel each other out in the case of the [tex]\(3^4\)[/tex].
5. Apply the Fourth Root:
- Simplify:
[tex]\[
\sqrt[4]{3^4 \cdot b} = \sqrt[4]{3^4} \cdot \sqrt[4]{b}
\][/tex]
6. Evaluate Further:
- [tex]\(\sqrt[4]{3^4}\)[/tex] simplifies to 3 (since taking the fourth root of [tex]\(3^4\)[/tex] returns the base 3);
[tex]\[
\sqrt[4]{3^4} = 3
\][/tex]
7. Combine the Results:
- Thus, the expression simplifies to:
[tex]\[
3 \cdot \sqrt[4]{b}
\][/tex]
8. Given Condition:
- If [tex]\(b = 1\)[/tex], then [tex]\(\sqrt[4]{1} = 1 \)[/tex].
9. Final Result:
- Substituting [tex]\(b = 1\)[/tex], the final result is:
[tex]\[
3 \times 1 = 3
\][/tex]
So, the fourth root of [tex]\(3^4 b^1\)[/tex] is:
[tex]\[
\sqrt[4]{3^4 b^1} = 3
\][/tex]
