Certainly! Here is a detailed, step-by-step solution for multiplying [tex]\(\left(\sqrt{2 x^3} + \sqrt{12 x}\right)\left(2 \sqrt{10 x^5} + \sqrt{6 x^2}\right)\)[/tex]:
1. Distribute Each Term:
We apply the distributive property: [tex]\((a+b)(c+d) = ac + ad + bc + bd\)[/tex].
Let [tex]\(a = \sqrt{2 x^3}\)[/tex], [tex]\(b = \sqrt{12 x}\)[/tex], [tex]\(c = 2 \sqrt{10 x^5}\)[/tex], and [tex]\(d = \sqrt{6 x^2}\)[/tex]. Then:
[tex]\[
(\sqrt{2 x^3} + \sqrt{12 x})(2 \sqrt{10 x^5} + \sqrt{6 x^2}) = (\sqrt{2 x^3})(2 \sqrt{10 x^5}) + (\sqrt{2 x^3})(\sqrt{6 x^2}) + (\sqrt{12 x})(2 \sqrt{10 x^5}) + (\sqrt{12 x})(\sqrt{6 x^2})
\][/tex]
2. Compute Each Term Separately:
[tex]\[
(\sqrt{2 x^3})(2 \sqrt{10 x^5})
\][/tex]
Simplify inside the square roots and combine:
[tex]\[
= 2 \sqrt{2 \cdot 10 x^3 \cdot x^5}
= 2 \sqrt{20 x^8}
= 2 \sqrt{4 \cdot 5 x^8}
= 2 \cdot 2 \sqrt{5 x^8}
= 4 x^4 \sqrt{5}
\][/tex]
[tex]\[
(\sqrt{2 x^3})(\sqrt{6 x^2})
\][/tex]
Combine inside the square roots:
[tex]\[
= \sqrt{2 \cdot 6 x^3 \cdot x^2}
= \sqrt{12 x^5}
= \sqrt{4 \cdot 3 x^5}
= 2 \sqrt{3 x^5}
\][/tex]
[tex]\[
(\sqrt{12 x})(2 \sqrt{10 x^5})
\][/tex]
Simplify inside the square roots and combine:
[tex]\[
= 2 \sqrt{12 \cdot 10 x \cdot x^5}
= 2 \sqrt{120 x^6}
= 2 \sqrt{4 \cdot 30 x^6}
= 2 \cdot 2 \sqrt{30 x^6}
= 4 x^3 \sqrt{30}
\][/tex]
[tex]\[
(\sqrt{12 x})(\sqrt{6 x^2})
\][/tex]
Combine inside the square roots:
[tex]\[
= \sqrt{12 \cdot 6 x \cdot x^2}
= \sqrt{72 x^3}
= \sqrt{36 \cdot 2 x^3}
= 6 \sqrt{2 x^3}
\][/tex]
3. Combine All Terms:
[tex]\[
= 4 x^4 \sqrt{5} + 2 \sqrt{3 x^5} + 4 x^3 \sqrt{30} + 6 \sqrt{2 x^3}
\][/tex]
So the final answer, broken down and combined, is:
[tex]\[
\left(\sqrt{2 x^3} + \sqrt{12 x}\right)\left(2 \sqrt{10 x^5} + \sqrt{6 x^2}\right) = 4 x^4 \sqrt{5} + 2 \sqrt{3 x^5} + 4 x^3 \sqrt{30} + 6 \sqrt{2 x^3}
\][/tex]
