Simplify ( square root of 2x^3+ square root of 12x)(2 square root of 10x^5+ square root of 6x^2)

$(\sqrt{2 x^{3}} + \sqrt{12 x}) (2 \sqrt{10 x^{5}} + \sqrt{6 x^{2}})$

Simplify terms.

Simplify each term.

Rewrite $2 x^{3}$ as $x^{2} \cdot (2 x)$ .

Factor out $x^{2}$ .

$(\sqrt{2 (x^{2} x)} + \sqrt{12 x}) (2 \sqrt{10 x^{5}} + \sqrt{6 x^{2}})$

Reorder $2$ and $x^{2}$ .

$(\sqrt{x^{2} \cdot 2 x} + \sqrt{12 x}) (2 \sqrt{10 x^{5}} + \sqrt{6 x^{2}})$

Add parentheses.

$(\sqrt{x^{2} \cdot (2 x)} + \sqrt{12 x}) (2 \sqrt{10 x^{5}} + \sqrt{6 x^{2}})$

Pull terms out from under the radical.

$(x \sqrt{2 x} + \sqrt{12 x}) (2 \sqrt{10 x^{5}} + \sqrt{6 x^{2}})$

Rewrite $12 x$ as $2^{2} \cdot (3 x)$ .

Factor $4$ out of $12$ .

$(x \sqrt{2 x} + \sqrt{4 (3) x}) (2 \sqrt{10 x^{5}} + \sqrt{6 x^{2}})$

Rewrite $4$ as $2^{2}$ .

$(x \sqrt{2 x} + \sqrt{2^{2} \cdot 3 x}) (2 \sqrt{10 x^{5}} + \sqrt{6 x^{2}})$

Add parentheses.

$(x \sqrt{2 x} + \sqrt{2^{2} \cdot (3 x)}) (2 \sqrt{10 x^{5}} + \sqrt{6 x^{2}})$

Pull terms out from under the radical.

$(x \sqrt{2 x} + 2 \sqrt{3 x}) (2 \sqrt{10 x^{5}} + \sqrt{6 x^{2}})$

Simplify each term.

Rewrite $10 x^{5}$ as ${(x^{2})}^{2} \cdot (10 x)$ .

Factor out $x^{4}$ .

$(x \sqrt{2 x} + 2 \sqrt{3 x}) (2 \sqrt{10 (x^{4} x)} + \sqrt{6 x^{2}})$

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$(x \sqrt{2 x} + 2 \sqrt{3 x}) (2 \sqrt{10 ({(x^{2})}^{2} x)} + \sqrt{6 x^{2}})$

Reorder $10$ and ${(x^{2})}^{2}$ .

$(x \sqrt{2 x} + 2 \sqrt{3 x}) (2 \sqrt{{(x^{2})}^{2} \cdot 10 x} + \sqrt{6 x^{2}})$

Add parentheses.

$(x \sqrt{2 x} + 2 \sqrt{3 x}) (2 \sqrt{{(x^{2})}^{2} \cdot (10 x)} + \sqrt{6 x^{2}})$

Pull terms out from under the radical.

$(x \sqrt{2 x} + 2 \sqrt{3 x}) (2 (x^{2} \sqrt{10 x}) + \sqrt{6 x^{2}})$

Reorder $6$ and $x^{2}$ .

$(x \sqrt{2 x} + 2 \sqrt{3 x}) (2 x^{2} \sqrt{10 x} + \sqrt{x^{2} \cdot 6})$

Pull terms out from under the radical.

$(x \sqrt{2 x} + 2 \sqrt{3 x}) (2 x^{2} \sqrt{10 x} + x \sqrt{6})$

Expand $(x \sqrt{2 x} + 2 \sqrt{3 x}) (2 x^{2} \sqrt{10 x} + x \sqrt{6})$ using the FOIL Method.

Apply the distributive property.

$x \sqrt{2 x} (2 x^{2} \sqrt{10 x} + x \sqrt{6}) + 2 \sqrt{3 x} (2 x^{2} \sqrt{10 x} + x \sqrt{6})$

Apply the distributive property.

$x \sqrt{2 x} (2 x^{2} \sqrt{10 x}) + x \sqrt{2 x} (x \sqrt{6}) + 2 \sqrt{3 x} (2 x^{2} \sqrt{10 x} + x \sqrt{6})$

Apply the distributive property.

$x \sqrt{2 x} (2 x^{2} \sqrt{10 x}) + x \sqrt{2 x} (x \sqrt{6}) + 2 \sqrt{3 x} (2 x^{2} \sqrt{10 x}) + 2 \sqrt{3 x} (x \sqrt{6})$

Simplify each term.

Rewrite using the commutative property of multiplication.

$2 x \sqrt{2 x} (x^{2} \sqrt{10 x}) + x \sqrt{2 x} (x \sqrt{6}) + 2 \sqrt{3 x} (2 x^{2} \sqrt{10 x}) + 2 \sqrt{3 x} (x \sqrt{6})$

Multiply $x$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$2 (x^{2} x) \sqrt{2 x} \sqrt{10 x} + x \sqrt{2 x} (x \sqrt{6}) + 2 \sqrt{3 x} (2 x^{2} \sqrt{10 x}) + 2 \sqrt{3 x} (x \sqrt{6})$

Multiply $x^{2}$ by $x$ .

Raise $x$ to the power of $1$ .

$2 (x^{2} x^{1}) \sqrt{2 x} \sqrt{10 x} + x \sqrt{2 x} (x \sqrt{6}) + 2 \sqrt{3 x} (2 x^{2} \sqrt{10 x}) + 2 \sqrt{3 x} (x \sqrt{6})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 x^{2 + 1} \sqrt{2 x} \sqrt{10 x} + x \sqrt{2 x} (x \sqrt{6}) + 2 \sqrt{3 x} (2 x^{2} \sqrt{10 x}) + 2 \sqrt{3 x} (x \sqrt{6})$

Add $2$ and $1$ .

$2 x^{3} \sqrt{2 x} \sqrt{10 x} + x \sqrt{2 x} (x \sqrt{6}) + 2 \sqrt{3 x} (2 x^{2} \sqrt{10 x}) + 2 \sqrt{3 x} (x \sqrt{6})$

Multiply $2 x^{3} \sqrt{2 x} \sqrt{10 x}$ .

Combine using the product rule for radicals.

$2 x^{3} \sqrt{10 x (2 x)} + x \sqrt{2 x} (x \sqrt{6}) + 2 \sqrt{3 x} (2 x^{2} \sqrt{10 x}) + 2 \sqrt{3 x} (x \sqrt{6})$

Multiply $2$ by $10$ .

$2 x^{3} \sqrt{20 x \cdot x} + x \sqrt{2 x} (x \sqrt{6}) + 2 \sqrt{3 x} (2 x^{2} \sqrt{10 x}) + 2 \sqrt{3 x} (x \sqrt{6})$

Raise $x$ to the power of $1$ .

$2 x^{3} \sqrt{20 (x^{1} x)} + x \sqrt{2 x} (x \sqrt{6}) + 2 \sqrt{3 x} (2 x^{2} \sqrt{10 x}) + 2 \sqrt{3 x} (x \sqrt{6})$

Raise $x$ to the power of $1$ .

$2 x^{3} \sqrt{20 (x^{1} x^{1})} + x \sqrt{2 x} (x \sqrt{6}) + 2 \sqrt{3 x} (2 x^{2} \sqrt{10 x}) + 2 \sqrt{3 x} (x \sqrt{6})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 x^{3} \sqrt{20 x^{1 + 1}} + x \sqrt{2 x} (x \sqrt{6}) + 2 \sqrt{3 x} (2 x^{2} \sqrt{10 x}) + 2 \sqrt{3 x} (x \sqrt{6})$

Add $1$ and $1$ .

$2 x^{3} \sqrt{20 x^{2}} + x \sqrt{2 x} (x \sqrt{6}) + 2 \sqrt{3 x} (2 x^{2} \sqrt{10 x}) + 2 \sqrt{3 x} (x \sqrt{6})$

Rewrite $20 x^{2}$ as ${(2 x)}^{2} \cdot 5$ .

Factor $4$ out of $20$ .

$2 x^{3} \sqrt{4 (5) x^{2}} + x \sqrt{2 x} (x \sqrt{6}) + 2 \sqrt{3 x} (2 x^{2} \sqrt{10 x}) + 2 \sqrt{3 x} (x \sqrt{6})$

Rewrite $4$ as $2^{2}$ .

$2 x^{3} \sqrt{2^{2} \cdot 5 x^{2}} + x \sqrt{2 x} (x \sqrt{6}) + 2 \sqrt{3 x} (2 x^{2} \sqrt{10 x}) + 2 \sqrt{3 x} (x \sqrt{6})$

Move $5$ .

$2 x^{3} \sqrt{2^{2} x^{2} \cdot 5} + x \sqrt{2 x} (x \sqrt{6}) + 2 \sqrt{3 x} (2 x^{2} \sqrt{10 x}) + 2 \sqrt{3 x} (x \sqrt{6})$

Rewrite $2^{2} x^{2}$ as ${(2 x)}^{2}$ .

$2 x^{3} \sqrt{{(2 x)}^{2} \cdot 5} + x \sqrt{2 x} (x \sqrt{6}) + 2 \sqrt{3 x} (2 x^{2} \sqrt{10 x}) + 2 \sqrt{3 x} (x \sqrt{6})$

Pull terms out from under the radical.

$2 x^{3} (2 x \sqrt{5}) + x \sqrt{2 x} (x \sqrt{6}) + 2 \sqrt{3 x} (2 x^{2} \sqrt{10 x}) + 2 \sqrt{3 x} (x \sqrt{6})$

Multiply $x^{3}$ by $x$ by adding the exponents.

Move $x$ .

$2 (x \cdot x^{3}) (2 \sqrt{5}) + x \sqrt{2 x} (x \sqrt{6}) + 2 \sqrt{3 x} (2 x^{2} \sqrt{10 x}) + 2 \sqrt{3 x} (x \sqrt{6})$

Multiply $x$ by $x^{3}$ .

Raise $x$ to the power of $1$ .

$2 (x^{1} x^{3}) (2 \sqrt{5}) + x \sqrt{2 x} (x \sqrt{6}) + 2 \sqrt{3 x} (2 x^{2} \sqrt{10 x}) + 2 \sqrt{3 x} (x \sqrt{6})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 x^{1 + 3} (2 \sqrt{5}) + x \sqrt{2 x} (x \sqrt{6}) + 2 \sqrt{3 x} (2 x^{2} \sqrt{10 x}) + 2 \sqrt{3 x} (x \sqrt{6})$

Add $1$ and $3$ .

$2 x^{4} (2 \sqrt{5}) + x \sqrt{2 x} (x \sqrt{6}) + 2 \sqrt{3 x} (2 x^{2} \sqrt{10 x}) + 2 \sqrt{3 x} (x \sqrt{6})$

Multiply $2$ by $2$ .

$4 x^{4} \sqrt{5} + x \sqrt{2 x} (x \sqrt{6}) + 2 \sqrt{3 x} (2 x^{2} \sqrt{10 x}) + 2 \sqrt{3 x} (x \sqrt{6})$

Rewrite using the commutative property of multiplication.

$4 x^{4} \sqrt{5} + \sqrt{6} x \sqrt{2 x} x + 2 \sqrt{3 x} (2 x^{2} \sqrt{10 x}) + 2 \sqrt{3 x} (x \sqrt{6})$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$4 x^{4} \sqrt{5} + \sqrt{6} (x \cdot x) \sqrt{2 x} + 2 \sqrt{3 x} (2 x^{2} \sqrt{10 x}) + 2 \sqrt{3 x} (x \sqrt{6})$

Multiply $x$ by $x$ .

$4 x^{4} \sqrt{5} + \sqrt{6} x^{2} \sqrt{2 x} + 2 \sqrt{3 x} (2 x^{2} \sqrt{10 x}) + 2 \sqrt{3 x} (x \sqrt{6})$

Multiply $\sqrt{6} x^{2} \sqrt{2 x}$ .

Combine using the product rule for radicals.

$4 x^{4} \sqrt{5} + x^{2} \sqrt{6 (2 x)} + 2 \sqrt{3 x} (2 x^{2} \sqrt{10 x}) + 2 \sqrt{3 x} (x \sqrt{6})$

Multiply $6$ by $2$ .

$4 x^{4} \sqrt{5} + x^{2} \sqrt{12 x} + 2 \sqrt{3 x} (2 x^{2} \sqrt{10 x}) + 2 \sqrt{3 x} (x \sqrt{6})$

Rewrite $12 x$ as $2^{2} \cdot (3 x)$ .

Factor $4$ out of $12$ .

$4 x^{4} \sqrt{5} + x^{2} \sqrt{4 (3) x} + 2 \sqrt{3 x} (2 x^{2} \sqrt{10 x}) + 2 \sqrt{3 x} (x \sqrt{6})$

Rewrite $4$ as $2^{2}$ .

$4 x^{4} \sqrt{5} + x^{2} \sqrt{2^{2} \cdot 3 x} + 2 \sqrt{3 x} (2 x^{2} \sqrt{10 x}) + 2 \sqrt{3 x} (x \sqrt{6})$

Add parentheses.

$4 x^{4} \sqrt{5} + x^{2} \sqrt{2^{2} \cdot (3 x)} + 2 \sqrt{3 x} (2 x^{2} \sqrt{10 x}) + 2 \sqrt{3 x} (x \sqrt{6})$

Pull terms out from under the radical.

$4 x^{4} \sqrt{5} + x^{2} (2 \sqrt{3 x}) + 2 \sqrt{3 x} (2 x^{2} \sqrt{10 x}) + 2 \sqrt{3 x} (x \sqrt{6})$

Rewrite using the commutative property of multiplication.

$4 x^{4} \sqrt{5} + 2 x^{2} \sqrt{3 x} + 2 \sqrt{3 x} (2 x^{2} \sqrt{10 x}) + 2 \sqrt{3 x} (x \sqrt{6})$

Multiply $2 \sqrt{3 x} (2 x^{2} \sqrt{10 x})$ .

Multiply $2$ by $2$ .

$4 x^{4} \sqrt{5} + 2 x^{2} \sqrt{3 x} + 4 \sqrt{3 x} (x^{2} \sqrt{10 x}) + 2 \sqrt{3 x} (x \sqrt{6})$

Combine using the product rule for radicals.

$4 x^{4} \sqrt{5} + 2 x^{2} \sqrt{3 x} + 4 (x^{2} \sqrt{3 x (10 x)}) + 2 \sqrt{3 x} (x \sqrt{6})$

Multiply $10$ by $3$ .

$4 x^{4} \sqrt{5} + 2 x^{2} \sqrt{3 x} + 4 (x^{2} \sqrt{30 x \cdot x}) + 2 \sqrt{3 x} (x \sqrt{6})$

Raise $x$ to the power of $1$ .

$4 x^{4} \sqrt{5} + 2 x^{2} \sqrt{3 x} + 4 (x^{2} \sqrt{30 (x^{1} x)}) + 2 \sqrt{3 x} (x \sqrt{6})$

Raise $x$ to the power of $1$ .

$4 x^{4} \sqrt{5} + 2 x^{2} \sqrt{3 x} + 4 (x^{2} \sqrt{30 (x^{1} x^{1})}) + 2 \sqrt{3 x} (x \sqrt{6})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$4 x^{4} \sqrt{5} + 2 x^{2} \sqrt{3 x} + 4 (x^{2} \sqrt{30 x^{1 + 1}}) + 2 \sqrt{3 x} (x \sqrt{6})$

Add $1$ and $1$ .

$4 x^{4} \sqrt{5} + 2 x^{2} \sqrt{3 x} + 4 (x^{2} \sqrt{30 x^{2}}) + 2 \sqrt{3 x} (x \sqrt{6})$

Reorder $30$ and $x^{2}$ .

$4 x^{4} \sqrt{5} + 2 x^{2} \sqrt{3 x} + 4 (x^{2} \sqrt{x^{2} \cdot 30}) + 2 \sqrt{3 x} (x \sqrt{6})$

Pull terms out from under the radical.

$4 x^{4} \sqrt{5} + 2 x^{2} \sqrt{3 x} + 4 (x^{2} (x \sqrt{30})) + 2 \sqrt{3 x} (x \sqrt{6})$

Multiply $x^{2}$ by $x$ by adding the exponents.

Move $x$ .

$4 x^{4} \sqrt{5} + 2 x^{2} \sqrt{3 x} + 4 (x \cdot x^{2} \sqrt{30}) + 2 \sqrt{3 x} (x \sqrt{6})$

Multiply $x$ by $x^{2}$ .

Raise $x$ to the power of $1$ .

$4 x^{4} \sqrt{5} + 2 x^{2} \sqrt{3 x} + 4 (x^{1} x^{2} \sqrt{30}) + 2 \sqrt{3 x} (x \sqrt{6})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$4 x^{4} \sqrt{5} + 2 x^{2} \sqrt{3 x} + 4 (x^{1 + 2} \sqrt{30}) + 2 \sqrt{3 x} (x \sqrt{6})$

Add $1$ and $2$ .

$4 x^{4} \sqrt{5} + 2 x^{2} \sqrt{3 x} + 4 (x^{3} \sqrt{30}) + 2 \sqrt{3 x} (x \sqrt{6})$

Multiply $2 \sqrt{3 x} (x \sqrt{6})$ .

Combine using the product rule for radicals.

$4 x^{4} \sqrt{5} + 2 x^{2} \sqrt{3 x} + 4 x^{3} \sqrt{30} + 2 (x \sqrt{3 x \cdot 6})$

Multiply $6$ by $3$ .

$4 x^{4} \sqrt{5} + 2 x^{2} \sqrt{3 x} + 4 x^{3} \sqrt{30} + 2 (x \sqrt{18 x})$

Rewrite $18 x$ as $3^{2} \cdot (2 x)$ .

Factor $9$ out of $18$ .

$4 x^{4} \sqrt{5} + 2 x^{2} \sqrt{3 x} + 4 x^{3} \sqrt{30} + 2 (x \sqrt{9 (2) x})$

Rewrite $9$ as $3^{2}$ .

$4 x^{4} \sqrt{5} + 2 x^{2} \sqrt{3 x} + 4 x^{3} \sqrt{30} + 2 (x \sqrt{3^{2} \cdot 2 x})$

Add parentheses.

$4 x^{4} \sqrt{5} + 2 x^{2} \sqrt{3 x} + 4 x^{3} \sqrt{30} + 2 (x \sqrt{3^{2} \cdot (2 x)})$

Pull terms out from under the radical.

$4 x^{4} \sqrt{5} + 2 x^{2} \sqrt{3 x} + 4 x^{3} \sqrt{30} + 2 (x (3 \sqrt{2 x}))$

Rewrite using the commutative property of multiplication.

$4 x^{4} \sqrt{5} + 2 x^{2} \sqrt{3 x} + 4 x^{3} \sqrt{30} + 2 (3 x \sqrt{2 x})$

Multiply $3$ by $2$ .

$4 x^{4} \sqrt{5} + 2 x^{2} \sqrt{3 x} + 4 x^{3} \sqrt{30} + 6 x \sqrt{2 x}$

Simplify ( square root of 2x^3+ square root of 12x)(2 square root of 10x^5+ square root of 6x^2)

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