To multiply the radicals together, we first use the property of square roots that states:
[tex]\[
\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}
\][/tex]
Let’s apply this to the given expression:
[tex]\[
\sqrt{12} \cdot \sqrt{6} \cdot \sqrt{x^3} \cdot \sqrt{x^7}
\][/tex]
First, we combine the numerical terms under one square root:
[tex]\[
\sqrt{12} \cdot \sqrt{6} = \sqrt{12 \cdot 6}
\][/tex]
Multiply the numbers inside the square root:
[tex]\[
12 \cdot 6 = 72
\][/tex]
So,
[tex]\[
\sqrt{12} \cdot \sqrt{6} = \sqrt{72}
\][/tex]
Next, we combine the variable terms:
[tex]\[
\sqrt{x^3} \cdot \sqrt{x^7} = \sqrt{x^3 \cdot x^7}
\][/tex]
Using the property of exponents that states [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex]:
[tex]\[
x^3 \cdot x^7 = x^{3+7} = x^{10}
\][/tex]
Thus,
[tex]\[
\sqrt{x^3} \cdot \sqrt{x^7} = \sqrt{x^{10}}
\][/tex]
Combine all terms under one square root:
[tex]\[
\sqrt{12} \cdot \sqrt{6} \cdot \sqrt{x^3} \cdot \sqrt{x^7} = \sqrt{72} \cdot \sqrt{x^{10}} = \sqrt{72 \cdot x^{10}}
\][/tex]
So the final expression is:
[tex]\[
\sqrt{72 x^{10}}
\][/tex]
Among the given options, the correct choice is:
(D) [tex]\(\sqrt{72 x^{10}}\)[/tex]
