Simplify (2x- square root of 3)(1+ square root of 3x^3)

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

Step 1

Simplify each term.

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

Factor out $x^{2}$ .

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

Reorder $3$ and $x^{2}$ .

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

Add parentheses.

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

Pull terms out from under the radical.

$(2 x - \sqrt{3}) (1 + x \sqrt{3 x})$

Step 2

Expand $(2 x - \sqrt{3}) (1 + x \sqrt{3 x})$ using the FOIL Method.

Apply the distributive property.

$2 x (1 + x \sqrt{3 x}) - \sqrt{3} (1 + x \sqrt{3 x})$

Apply the distributive property.

$2 x \cdot 1 + 2 x (x \sqrt{3 x}) - \sqrt{3} (1 + x \sqrt{3 x})$

Apply the distributive property.

$2 x \cdot 1 + 2 x (x \sqrt{3 x}) - \sqrt{3} \cdot 1 - \sqrt{3} (x \sqrt{3 x})$

Step 3

Simplify each term.

Multiply $2$ by $1$ .

$2 x + 2 x (x \sqrt{3 x}) - \sqrt{3} \cdot 1 - \sqrt{3} (x \sqrt{3 x})$

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

Move $x$ .

$2 x + 2 (x \cdot x) \sqrt{3 x} - \sqrt{3} \cdot 1 - \sqrt{3} (x \sqrt{3 x})$

Multiply $x$ by $x$ .

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

Multiply $- 1$ by $1$ .

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

Multiply $- \sqrt{3} (x \sqrt{3 x})$ .

Combine using the product rule for radicals.

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

Multiply $3$ by $3$ .

$2 x + 2 x^{2} \sqrt{3 x} - \sqrt{3} - 1 (x \sqrt{9 x})$

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

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

Pull terms out from under the radical.

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

Rewrite using the commutative property of multiplication.

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

Multiply $3$ by $- 1$ .

$2 x + 2 x^{2} \sqrt{3 x} - \sqrt{3} - 3 x \sqrt{x}$