To simplify the expression [tex]\(\sqrt{\sqrt[3]{x^{10} \cdot x^2}}\)[/tex], we can proceed step-by-step as follows:
1. Combine the exponents inside the cube root:
Start by simplifying the product inside the cube root. We have:
[tex]\[
x^{10} \cdot x^2
\][/tex]
Since the bases are the same (both [tex]\(x\)[/tex]), we can add the exponents:
[tex]\[
x^{10} \cdot x^2 = x^{10 + 2} = x^{12}
\][/tex]
2. Take the cube root:
Next, take the cube root of [tex]\(x^{12}\)[/tex]. The cube root of a power can be found by dividing the exponent by 3:
[tex]\[
\sqrt[3]{x^{12}} = x^{12 / 3} = x^4
\][/tex]
3. Take the square root:
Finally, take the square root of [tex]\(x^4\)[/tex]. The square root of a power can be found by dividing the exponent by 2:
[tex]\[
\sqrt{x^4} = x^{4 / 2} = x^2
\][/tex]
Therefore, the final simplified expression is:
[tex]\[
\sqrt{\sqrt[3]{x^{10} \cdot x^2}} = x^2
\][/tex]
