Multiply: [tex]\left(\sqrt{2x^3} + \sqrt{12x}\right)\left(2\sqrt{10x^5} + \sqrt{6x^2}\right)[/tex]

[tex]\ \textless \ br/\ \textgreater \ \begin{array}{l}\ \textless \ br/\ \textgreater \ 2 \sqrt{10x^4} + 2 \sqrt{3x^3} + 4 \sqrt{15x^3} + 6 \sqrt{2x} \\\ \textless \ br/\ \textgreater \ 2x^2 \sqrt{5} + 2x \sqrt{3x} + 2x^3 \sqrt{30} + 3x \sqrt{2x} \\\ \textless \ br/\ \textgreater \ 4x^4 \sqrt{5} + 2x^2 \sqrt{3x} + 4x^3 \sqrt{30} + 6x \sqrt{2x} \\\ \textless \ br/\ \textgreater \ x^4 \sqrt{20} + x^2 \sqrt{6x} + x^3 \sqrt{120} + x \sqrt{12x}\ \textless \ br/\ \textgreater \ \end{array}\ \textless \ br/\ \textgreater \ [/tex]



Answer :

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]