To simplify the expression [tex]\(\sqrt{\sqrt[3]{x^{10} \cdot x^2}}\)[/tex], we can follow a step-by-step process:
1. Combine the exponents inside the innermost part of the expression:
When multiplying expressions with the same base, you add the exponents. Therefore:
[tex]\[
x^{10} \cdot x^2 = x^{10+2} = x^{12}
\][/tex]
2. Take the cube root of [tex]\(x^{12}\)[/tex]:
Taking the cube root of an expression can be written as raising the expression to the power of [tex]\(\frac{1}{3}\)[/tex]. Hence:
[tex]\[
\sqrt[3]{x^{12}} = \left(x^{12}\right)^{\frac{1}{3}}
\][/tex]
Using the property of exponents, [tex]\((a^m)^n = a^{m \cdot n}\)[/tex], we get:
[tex]\[
\left(x^{12}\right)^{\frac{1}{3}} = x^{12 \cdot \frac{1}{3}} = x^4
\][/tex]
3. Take the square root of [tex]\(x^4\)[/tex]:
Taking the square root of an expression means raising it to the power of [tex]\(\frac{1}{2}\)[/tex]. Therefore:
[tex]\[
\sqrt{x^4} = \left(x^4\right)^{\frac{1}{2}}
\][/tex]
Applying the property of exponents again:
[tex]\[
\left(x^4\right)^{\frac{1}{2}} = x^{4 \cdot \frac{1}{2}} = x^2
\][/tex]
So, the simplified expression is:
[tex]\[
\sqrt{\sqrt[3]{x^{10} \cdot x^2}} = x^2
\][/tex]
