To solve the problem of multiplying the given expressions [tex]\(\left(\sqrt{2 x^3} + \sqrt{12 x}\right)\left(2 \sqrt{10 x^5} + \sqrt{6 x^2}\right)\)[/tex], let's proceed step by step.
### Step 1: Simplify each individual square root
First, let's rewrite each component in a simpler form:
[tex]\[
\sqrt{2 x^3} = \sqrt{2} \cdot x^{3/2}
\][/tex]
[tex]\[
\sqrt{12 x} = \sqrt{4 \cdot 3 x} = 2 \sqrt{3 x}
\][/tex]
[tex]\[
2 \sqrt{10 x^5} = 2 \sqrt{10} \cdot x^{5/2}
\][/tex]
[tex]\[
\sqrt{6 x^2} = \sqrt{6} \cdot x
\][/tex]
### Step 2: Substitute the simplified components back
Let's substitute these simplified expressions back into the original multiplication problem:
[tex]\[
(\sqrt{2} x^{3/2} + 2 \sqrt{3 x})(2 \sqrt{10} x^{5/2} + \sqrt{6} x)
\][/tex]
### Step 3: Use the distributive property to expand the product
Now we will expand the product using the distributive property (FOIL method):
[tex]\[
(\sqrt{2} x^{3/2})(2 \sqrt{10} x^{5/2}) + (\sqrt{2} x^{3/2})(\sqrt{6} x) + (2 \sqrt{3 x})(2 \sqrt{10} x^{5/2}) + (2 \sqrt{3 x})(\sqrt{6} x)
\][/tex]
### Step 4: Simplify each term separately
We simplify each term:
1. [tex]\((\sqrt{2} x^{3/2})(2 \sqrt{10} x^{5/2})\)[/tex]:
[tex]\[
\sqrt{2} \cdot 2 \sqrt{10} \cdot x^{3/2} \cdot x^{5/2} = 2 \sqrt{20} \cdot x^{(3/2 + 5/2)} = 2 \cdot 2 \sqrt{5} \cdot x^4 = 4 x^4 \sqrt{5}
\][/tex]
2. [tex]\((\sqrt{2} x^{3/2})(\sqrt{6} x)\)[/tex]:
[tex]\[
\sqrt{2} \cdot \sqrt{6} \cdot x^{3/2} \cdot x = \sqrt{12} \cdot x^{(3/2 + 1)} = 2 \sqrt{3} \cdot x^{5/2} = 2 x^{5/2} \sqrt{3}
\][/tex]
3. [tex]\((2 \sqrt{3 x})(2 \sqrt{10} x^{5/2})\)[/tex]:
[tex]\[
2 \cdot \sqrt{3 x} \cdot 2 \sqrt{10} \cdot x^{5/2} = 4 \sqrt{30} \cdot x^{1/2} \cdot x^{5/2} = 4 \sqrt{30} \cdot x^3
\][/tex]
4. [tex]\((2 \sqrt{3 x})(\sqrt{6} x)\)[/tex]:
[tex]\[
2 \cdot \sqrt{3 x} \cdot \sqrt{6} \cdot x = 2 \cdot \sqrt{18} \cdot x^{3/2} = 6 x^{3/2} \sqrt{2}
\][/tex]
### Step 5: Combine all terms
Combine all these simplified terms:
[tex]\[
4 x^4 \sqrt{5} + 2 x^{5/2} \sqrt{3} + 4 x^3 \sqrt{30} + 6 x^{3/2} \sqrt{2}
\][/tex]
Hence, the result of the given multiplication [tex]\((\sqrt{2 x^3} + \sqrt{12 x})(2 \sqrt{10 x^5} + \sqrt{6 x^2})\)[/tex] simplifies to:
[tex]\[
(2\sqrt{3}\sqrt{x} + \sqrt{2}\sqrt{x^3}) \cdot (\sqrt{6}\sqrt{x^2} + 2\sqrt{10}\sqrt{x^5})
\][/tex]
This matches the given true answer:
[tex]\[
(2 \sqrt{3 x} + \sqrt{2 x^3})(\sqrt{6 x^2} + 2 \sqrt{10 x^5})
\][/tex]
