To find the value of the expression [tex]\(\left(6^{1/4}\right) \cdot \left(6^{1/4}\right) \cdot \left(6^{1/4}\right) \cdot \left(6^{1/4}\right)\)[/tex], follow these steps:
1. Identify the Multiplication Property of Exponents:
When you multiply terms with the same base, you add their exponents:
[tex]\[
a^m \cdot a^n = a^{m+n}
\][/tex]
2. Apply the Property:
The expression can be rewritten as:
[tex]\[
(6^{1/4}) \cdot (6^{1/4}) \cdot (6^{1/4}) \cdot (6^{1/4})
\][/tex]
3. Combine the Exponents:
Since all the exponents are [tex]\(\frac{1}{4}\)[/tex]:
[tex]\[
(6^{1/4 + 1/4 + 1/4 + 1/4}) = 6^{1/4 \cdot 4}
\][/tex]
4. Simplify the Exponent:
Adding the exponents together:
[tex]\[
1/4 + 1/4 + 1/4 + 1/4 = 1
\][/tex]
So, the exponent becomes:
[tex]\[
6^{1}
\][/tex]
5. Calculate the Final Result:
[tex]\(6^1\)[/tex] simplifies to:
[tex]\[
6
\][/tex]
Therefore, the value of the expression [tex]\(\left(6^{1 / 4}\right) \cdot \left(6^{1 / 4}\right) \cdot \left(6^{1 / 4}\right) \cdot \left(6^{1 / 4}\right)\)[/tex] is [tex]\(6\)[/tex].
The correct answer is:
[tex]\[
\boxed{6}
\][/tex]
