Simplify (9-4 square root of 3y)(3+6 square root of 21x)

$(9 - 4 \sqrt{3 y}) (3 + 6 \sqrt{21 x})$

Step 1

Expand $(9 - 4 \sqrt{3 y}) (3 + 6 \sqrt{21 x})$ using the FOIL Method.

Apply the distributive property.

$9 (3 + 6 \sqrt{21 x}) - 4 \sqrt{3 y} (3 + 6 \sqrt{21 x})$

Apply the distributive property.

$9 \cdot 3 + 9 (6 \sqrt{21 x}) - 4 \sqrt{3 y} (3 + 6 \sqrt{21 x})$

Apply the distributive property.

$9 \cdot 3 + 9 (6 \sqrt{21 x}) - 4 \sqrt{3 y} \cdot 3 - 4 \sqrt{3 y} (6 \sqrt{21 x})$

Step 2

Simplify each term.

Multiply $9$ by $3$ .

$27 + 9 (6 \sqrt{21 x}) - 4 \sqrt{3 y} \cdot 3 - 4 \sqrt{3 y} (6 \sqrt{21 x})$

Multiply $6$ by $9$ .

$27 + 54 \sqrt{21 x} - 4 \sqrt{3 y} \cdot 3 - 4 \sqrt{3 y} (6 \sqrt{21 x})$

Multiply $3$ by $- 4$ .

$27 + 54 \sqrt{21 x} - 12 \sqrt{3 y} - 4 \sqrt{3 y} (6 \sqrt{21 x})$

Multiply $- 4 \sqrt{3 y} (6 \sqrt{21 x})$ .

Multiply $6$ by $- 4$ .

$27 + 54 \sqrt{21 x} - 12 \sqrt{3 y} - 24 \sqrt{3 y} \sqrt{21 x}$

Combine using the product rule for radicals.

$27 + 54 \sqrt{21 x} - 12 \sqrt{3 y} - 24 \sqrt{21 x (3 y)}$

Multiply $3$ by $21$ .

$27 + 54 \sqrt{21 x} - 12 \sqrt{3 y} - 24 \sqrt{63 x y}$

Rewrite $63 x y$ as $3^{2} \cdot (7 x y)$ .

Factor $9$ out of $63$ .

$27 + 54 \sqrt{21 x} - 12 \sqrt{3 y} - 24 \sqrt{9 (7) x y}$

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

$27 + 54 \sqrt{21 x} - 12 \sqrt{3 y} - 24 \sqrt{3^{2} \cdot 7 x y}$

Add parentheses.

$27 + 54 \sqrt{21 x} - 12 \sqrt{3 y} - 24 \sqrt{3^{2} \cdot 7 (x y)}$

Add parentheses.

$27 + 54 \sqrt{21 x} - 12 \sqrt{3 y} - 24 \sqrt{3^{2} \cdot (7 x y)}$

Pull terms out from under the radical.

$27 + 54 \sqrt{21 x} - 12 \sqrt{3 y} - 24 (3 \sqrt{7 x y})$

Multiply $3$ by $- 24$ .

$27 + 54 \sqrt{21 x} - 12 \sqrt{3 y} - 72 \sqrt{7 x y}$