Sure, let’s solve the given mathematical expression step-by-step.
We need to simplify the expression [tex]\(\sqrt{\sqrt[3]{x^{10} \cdot x^2}}\)[/tex].
Step 1: Simplify the inside of the cube root first.
[tex]\[ x^{10} \cdot x^2 \][/tex]
Using the properties of exponents, we can combine the [tex]\(x\)[/tex] terms:
[tex]\[ x^{10} \cdot x^2 = x^{10 + 2} = x^{12} \][/tex]
So the expression now becomes:
[tex]\[ \sqrt{\sqrt[3]{x^{12}}} \][/tex]
Step 2: Simplify the cube root.
[tex]\[ \sqrt[3]{x^{12}} \][/tex]
By applying the property of exponents, [tex]\(\sqrt[3]{x^{12}}\)[/tex] can be written as:
[tex]\[ (x^{12})^{1/3} \][/tex]
Now we raise the exponent to the power 1/3:
[tex]\[ (x^{12})^{1/3} = x^{12 \cdot (1/3)} = x^{4} \][/tex]
So the expression simplifies to:
[tex]\[ \sqrt{x^4} \][/tex]
Step 3: Simplify the square root.
[tex]\[ \sqrt{x^4} \][/tex]
Applying the property of exponents again, the square root can be written as:
[tex]\[ (x^4)^{1/2} \][/tex]
Now we raise the exponent to the power 1/2:
[tex]\[ (x^4)^{1/2} = x^{4 \cdot (1/2)} = x^2 \][/tex]
Therefore, the simplified form of the given expression is:
[tex]\[ \sqrt{\sqrt[3]{x^{10} \cdot x^2}} = (x^4)^{1/2} = x^2 \][/tex]
So, the final answer is:
[tex]\[ x^2 \][/tex]
