To solve the expression [tex]\(\sqrt[6]{\sqrt[4]{a^3}}\)[/tex], let's break it down into step-by-step parts:
1. Starting with the inner expression:
[tex]\[
\sqrt[4]{a^3}
\][/tex]
The fourth root of [tex]\(a^3\)[/tex] can be written as:
[tex]\[
(a^3)^{1/4}
\][/tex]
2. Next, we handle the outer expression:
[tex]\[
\sqrt[6]{\sqrt[4]{a^3}}
\][/tex]
We'll substitute the equivalent form of the inner expression from step 1:
[tex]\[
\sqrt[6]{(a^3)^{1/4}}
\][/tex]
3. Combining the exponents:
To simplify the sixth root of [tex]\((a^3)^{1/4}\)[/tex], we treat the entire expression as a single exponentiation process. The operation is:
[tex]\[
\left((a^3)^{1/4}\right)^{1/6}
\][/tex]
Using the property of exponents [tex]\((x^a)^b = x^{a \cdot b}\)[/tex], we can combine the exponents:
[tex]\[
(a^3)^{1/4 \cdot 1/6}
\][/tex]
4. Simplify the exponent:
Multiply the exponents [tex]\(\frac{1}{4}\)[/tex] and [tex]\(\frac{1}{6}\)[/tex]:
[tex]\[
\frac{1}{4} \cdot \frac{1}{6} = \frac{1}{24}
\][/tex]
Therefore, we have:
[tex]\[
(a^3)^{1/24}
\][/tex]
So, the original expression [tex]\(\sqrt[6]{\sqrt[4]{a^3}}\)[/tex] simplifies to:
[tex]\[
(a^3)^{1/24}
\][/tex]
Hence, the detailed step-by-step solution yields the final result:
[tex]\[
\boxed{(a^3)^{1/24}}
\][/tex]
