To solve the expression [tex]\(\sqrt{\sqrt[3]{x^{10} \cdot x^2}}\)[/tex], we need to simplify step by step, applying the rules of exponents and roots.
1. Combine the Exponents Inside the Cubic Root:
Start with the inside of the cubic root: [tex]\(x^{10} \cdot x^2\)[/tex]:
[tex]\[
x^{10} \cdot x^2 = x^{10 + 2} = x^{12}
\][/tex]
2. Apply the Cubic Root:
Next, take the cubic root of [tex]\(x^{12}\)[/tex]:
[tex]\[
\sqrt[3]{x^{12}} = (x^{12})^{1/3}
\][/tex]
By the rules of exponents, [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[
(x^{12})^{1/3} = x^{12 \cdot \frac{1}{3}} = x^{4}
\][/tex]
3. Apply the Square Root:
Finally, take the square root of the result:
[tex]\[
\sqrt{x^4} = (x^4)^{1/2}
\][/tex]
Using the exponent rule again:
[tex]\[
(x^4)^{1/2} = x^{4 \cdot \frac{1}{2}} = x^{2}
\][/tex]
4. Conclusion:
Therefore, the simplified expression is:
[tex]\[
\sqrt{\sqrt[3]{x^{10} \cdot x^2}} = x^{2}
\][/tex]
