To simplify the expression [tex]\(\sqrt[12]{\left(x^4\right)^{1 / 3}}\)[/tex], we can follow these steps:
1. Simplify the inner exponentiation:
[tex]\[
(x^4)^{1/3}
\][/tex]
We use the property of exponents which states [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]. Applying this property:
[tex]\[
(x^4)^{1/3} = x^{4 \cdot (1/3)} = x^{4/3}
\][/tex]
2. Simplify the outer root:
[tex]\[
\sqrt[12]{x^{4/3}}
\][/tex]
The 12th root of a number can be expressed as raising the number to the power of [tex]\(1/12\)[/tex]. Therefore:
[tex]\[
\sqrt[12]{x^{4/3}} = \left(x^{4/3}\right)^{1/12}
\][/tex]
3. Combine the exponents:
Again, using the property of exponents [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[
\left(x^{4/3}\right)^{1/12} = x^{(4/3) \cdot (1/12)}
\][/tex]
4. Multiply the exponents:
[tex]\[
x^{(4/3) \cdot (1/12)} = x^{4/(3 \cdot 12)} = x^{4/36} = x^{1/9}
\][/tex]
So, the simplified form of the expression [tex]\(\sqrt[12]{\left(x^4\right)^{1 / 3}}\)[/tex] is:
[tex]\[
x^{1/9}
\][/tex]
