Simplify 2 square root of 8x^3(3 square root of 10x^4-x square root of 5x^2)

$2 \sqrt{8 x^{3}} (3 \sqrt{10 x^{4}} - x \sqrt{5 x^{2}})$

Step 1

Rewrite $8 x^{3}$ as ${(2 x)}^{2} \cdot (2 x)$ .

Factor $4$ out of $8$ .

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

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

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

Factor out $x^{2}$ .

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

Move $2$ .

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

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

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

Add parentheses.

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

Step 2

Simplify by cancelling exponent with radical.

Pull terms out from under the radical.

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

Multiply $2$ by $2$ .

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

Step 3

Simplify each term.

Rewrite $10 x^{4}$ as ${(x^{2})}^{2} \cdot 10$ .

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

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

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

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

Pull terms out from under the radical.

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

Reorder $5$ and $x^{2}$ .

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

Pull terms out from under the radical.

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Step 4

Apply the distributive property.

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

Step 5

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

Move $x^{2}$ .

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

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

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

$4 (x^{2} x^{1}) \sqrt{2 x} (3 \sqrt{10}) + 4 x \sqrt{2 x} (- x^{2} \sqrt{5})$

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

$4 x^{2 + 1} \sqrt{2 x} (3 \sqrt{10}) + 4 x \sqrt{2 x} (- x^{2} \sqrt{5})$

Add $2$ and $1$ .

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

Step 6

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

Move $x^{2}$ .

$4 x^{3} \sqrt{2 x} (3 \sqrt{10}) + 4 (x^{2} x) \sqrt{2 x} (- 1 \sqrt{5})$

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

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

$4 x^{3} \sqrt{2 x} (3 \sqrt{10}) + 4 (x^{2} x^{1}) \sqrt{2 x} (- 1 \sqrt{5})$

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

$4 x^{3} \sqrt{2 x} (3 \sqrt{10}) + 4 x^{2 + 1} \sqrt{2 x} (- 1 \sqrt{5})$

Add $2$ and $1$ .

$4 x^{3} \sqrt{2 x} (3 \sqrt{10}) + 4 x^{3} \sqrt{2 x} (- 1 \sqrt{5})$

Step 7

Simplify each term.

Multiply $4 x^{3} \sqrt{2 x} (3 \sqrt{10})$ .

Multiply $3$ by $4$ .

$12 x^{3} \sqrt{2 x} \sqrt{10} + 4 x^{3} \sqrt{2 x} (- 1 \sqrt{5})$

Combine using the product rule for radicals.

$12 x^{3} \sqrt{10 (2 x)} + 4 x^{3} \sqrt{2 x} (- 1 \sqrt{5})$

Multiply $10$ by $2$ .

$12 x^{3} \sqrt{20 x} + 4 x^{3} \sqrt{2 x} (- 1 \sqrt{5})$

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

Factor $4$ out of $20$ .

$12 x^{3} \sqrt{4 (5) x} + 4 x^{3} \sqrt{2 x} (- 1 \sqrt{5})$

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

$12 x^{3} \sqrt{2^{2} \cdot 5 x} + 4 x^{3} \sqrt{2 x} (- 1 \sqrt{5})$

Add parentheses.

$12 x^{3} \sqrt{2^{2} \cdot (5 x)} + 4 x^{3} \sqrt{2 x} (- 1 \sqrt{5})$

Pull terms out from under the radical.

$12 x^{3} (2 \sqrt{5 x}) + 4 x^{3} \sqrt{2 x} (- 1 \sqrt{5})$

Multiply $2$ by $12$ .

$24 x^{3} \sqrt{5 x} + 4 x^{3} \sqrt{2 x} (- 1 \sqrt{5})$

Rewrite $- 1 \sqrt{5}$ as $- \sqrt{5}$ .

$24 x^{3} \sqrt{5 x} + 4 x^{3} \sqrt{2 x} (- \sqrt{5})$

Multiply $4 x^{3} \sqrt{2 x} (- \sqrt{5})$ .

Multiply $- 1$ by $4$ .

$24 x^{3} \sqrt{5 x} - 4 x^{3} \sqrt{2 x} \sqrt{5}$

Combine using the product rule for radicals.

$24 x^{3} \sqrt{5 x} - 4 x^{3} \sqrt{5 (2 x)}$

Multiply $5$ by $2$ .

$24 x^{3} \sqrt{5 x} - 4 x^{3} \sqrt{10 x}$

Simplify 2 square root of 8x^3(3 square root of 10x^4-x square root of 5x^2)

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