Simplify ( cube root of 9-2)( cube root of 81+2 cube root of 9+4)

$(\sqrt[3]{9} - 2) (\sqrt[3]{81} + 2 \sqrt[3]{9} + 4)$

Simplify each term.

Rewrite $81$ as $3^{3} \cdot 3$ .

Factor $27$ out of $81$ .

$(\sqrt[3]{9} - 2) (\sqrt[3]{27 (3)} + 2 \sqrt[3]{9} + 4)$

Rewrite $27$ as $3^{3}$ .

$(\sqrt[3]{9} - 2) (\sqrt[3]{3^{3} \cdot 3} + 2 \sqrt[3]{9} + 4)$

Pull terms out from under the radical.

$(\sqrt[3]{9} - 2) (3 \sqrt[3]{3} + 2 \sqrt[3]{9} + 4)$

Expand $(\sqrt[3]{9} - 2) (3 \sqrt[3]{3} + 2 \sqrt[3]{9} + 4)$ by multiplying each term in the first expression by each term in the second expression.

$\sqrt[3]{9} (3 \sqrt[3]{3}) + \sqrt[3]{9} (2 \sqrt[3]{9}) + \sqrt[3]{9} \cdot 4 - 2 (3 \sqrt[3]{3}) - 2 (2 \sqrt[3]{9}) - 2 \cdot 4$

Simplify terms.

Simplify each term.

Multiply $\sqrt[3]{9} (3 \sqrt[3]{3})$ .

Combine using the product rule for radicals.

$3 \sqrt[3]{9 \cdot 3} + \sqrt[3]{9} (2 \sqrt[3]{9}) + \sqrt[3]{9} \cdot 4 - 2 (3 \sqrt[3]{3}) - 2 (2 \sqrt[3]{9}) - 2 \cdot 4$

Multiply $9$ by $3$ .

$3 \sqrt[3]{27} + \sqrt[3]{9} (2 \sqrt[3]{9}) + \sqrt[3]{9} \cdot 4 - 2 (3 \sqrt[3]{3}) - 2 (2 \sqrt[3]{9}) - 2 \cdot 4$

Rewrite $27$ as $3^{3}$ .

$3 \sqrt[3]{3^{3}} + \sqrt[3]{9} (2 \sqrt[3]{9}) + \sqrt[3]{9} \cdot 4 - 2 (3 \sqrt[3]{3}) - 2 (2 \sqrt[3]{9}) - 2 \cdot 4$

Pull terms out from under the radical, assuming real numbers.

$3 \cdot 3 + \sqrt[3]{9} (2 \sqrt[3]{9}) + \sqrt[3]{9} \cdot 4 - 2 (3 \sqrt[3]{3}) - 2 (2 \sqrt[3]{9}) - 2 \cdot 4$

Multiply $3$ by $3$ .

$9 + \sqrt[3]{9} (2 \sqrt[3]{9}) + \sqrt[3]{9} \cdot 4 - 2 (3 \sqrt[3]{3}) - 2 (2 \sqrt[3]{9}) - 2 \cdot 4$

Multiply $\sqrt[3]{9} (2 \sqrt[3]{9})$ .

Raise $\sqrt[3]{9}$ to the power of $1$ .

$9 + 2 ({\sqrt[3]{9}}^{1} \sqrt[3]{9}) + \sqrt[3]{9} \cdot 4 - 2 (3 \sqrt[3]{3}) - 2 (2 \sqrt[3]{9}) - 2 \cdot 4$

Raise $\sqrt[3]{9}$ to the power of $1$ .

$9 + 2 ({\sqrt[3]{9}}^{1} {\sqrt[3]{9}}^{1}) + \sqrt[3]{9} \cdot 4 - 2 (3 \sqrt[3]{3}) - 2 (2 \sqrt[3]{9}) - 2 \cdot 4$

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

$9 + 2 {\sqrt[3]{9}}^{1 + 1} + \sqrt[3]{9} \cdot 4 - 2 (3 \sqrt[3]{3}) - 2 (2 \sqrt[3]{9}) - 2 \cdot 4$

Add $1$ and $1$ .

$9 + 2 {\sqrt[3]{9}}^{2} + \sqrt[3]{9} \cdot 4 - 2 (3 \sqrt[3]{3}) - 2 (2 \sqrt[3]{9}) - 2 \cdot 4$

Rewrite ${\sqrt[3]{9}}^{2}$ as ${(9^{2})}^{\frac{1}{3}}$ .

$9 + 2 \sqrt[3]{9^{2}} + \sqrt[3]{9} \cdot 4 - 2 (3 \sqrt[3]{3}) - 2 (2 \sqrt[3]{9}) - 2 \cdot 4$

Raise $9$ to the power of $2$ .

$9 + 2 \sqrt[3]{81} + \sqrt[3]{9} \cdot 4 - 2 (3 \sqrt[3]{3}) - 2 (2 \sqrt[3]{9}) - 2 \cdot 4$

Rewrite $81$ as $3^{3} \cdot 3$ .

Factor $27$ out of $81$ .

$9 + 2 \sqrt[3]{27 (3)} + \sqrt[3]{9} \cdot 4 - 2 (3 \sqrt[3]{3}) - 2 (2 \sqrt[3]{9}) - 2 \cdot 4$

Rewrite $27$ as $3^{3}$ .

$9 + 2 \sqrt[3]{3^{3} \cdot 3} + \sqrt[3]{9} \cdot 4 - 2 (3 \sqrt[3]{3}) - 2 (2 \sqrt[3]{9}) - 2 \cdot 4$

Pull terms out from under the radical.

$9 + 2 (3 \sqrt[3]{3}) + \sqrt[3]{9} \cdot 4 - 2 (3 \sqrt[3]{3}) - 2 (2 \sqrt[3]{9}) - 2 \cdot 4$

Multiply $3$ by $2$ .

$9 + 6 \sqrt[3]{3} + \sqrt[3]{9} \cdot 4 - 2 (3 \sqrt[3]{3}) - 2 (2 \sqrt[3]{9}) - 2 \cdot 4$

Move $4$ to the left of $\sqrt[3]{9}$ .

$9 + 6 \sqrt[3]{3} + 4 \cdot \sqrt[3]{9} - 2 (3 \sqrt[3]{3}) - 2 (2 \sqrt[3]{9}) - 2 \cdot 4$

Multiply $3$ by $- 2$ .

$9 + 6 \sqrt[3]{3} + 4 \sqrt[3]{9} - 6 \sqrt[3]{3} - 2 (2 \sqrt[3]{9}) - 2 \cdot 4$

Multiply $2$ by $- 2$ .

$9 + 6 \sqrt[3]{3} + 4 \sqrt[3]{9} - 6 \sqrt[3]{3} - 4 \sqrt[3]{9} - 2 \cdot 4$

Multiply $- 2$ by $4$ .

$9 + 6 \sqrt[3]{3} + 4 \sqrt[3]{9} - 6 \sqrt[3]{3} - 4 \sqrt[3]{9} - 8$

Simplify by adding terms.

Subtract $8$ from $9$ .

$1 + 6 \sqrt[3]{3} + 4 \sqrt[3]{9} - 6 \sqrt[3]{3} - 4 \sqrt[3]{9}$

Subtract $6 \sqrt[3]{3}$ from $6 \sqrt[3]{3}$ .

$1 + 0 + 4 \sqrt[3]{9} - 4 \sqrt[3]{9}$

Add $1$ and $0$ .

$1 + 4 \sqrt[3]{9} - 4 \sqrt[3]{9}$

Subtract $4 \sqrt[3]{9}$ from $4 \sqrt[3]{9}$ .

$1 + 0$

Add $1$ and $0$ .

$1$

Simplify ( cube root of 9-2)( cube root of 81+2 cube root of 9+4)

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