Simplify ( cube root of 8+ cube root of 4)( cube root of 64- cube root of 32+ cube root of 16)

$(\sqrt[3]{8} + \sqrt[3]{4}) (\sqrt[3]{64} - \sqrt[3]{32} + \sqrt[3]{16})$

Step 1

Simplify terms.

Simplify each term.

Rewrite $8$ as $2^{3}$ .

$(\sqrt[3]{2^{3}} + \sqrt[3]{4}) (\sqrt[3]{64} - \sqrt[3]{32} + \sqrt[3]{16})$

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

$(2 + \sqrt[3]{4}) (\sqrt[3]{64} - \sqrt[3]{32} + \sqrt[3]{16})$

Simplify each term.

Rewrite $64$ as $4^{3}$ .

$(2 + \sqrt[3]{4}) (\sqrt[3]{4^{3}} - \sqrt[3]{32} + \sqrt[3]{16})$

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

$(2 + \sqrt[3]{4}) (4 - \sqrt[3]{32} + \sqrt[3]{16})$

Rewrite $32$ as $2^{3} \cdot 4$ .

Factor $8$ out of $32$ .

$(2 + \sqrt[3]{4}) (4 - \sqrt[3]{8 (4)} + \sqrt[3]{16})$

Rewrite $8$ as $2^{3}$ .

$(2 + \sqrt[3]{4}) (4 - \sqrt[3]{2^{3} \cdot 4} + \sqrt[3]{16})$

Pull terms out from under the radical.

$(2 + \sqrt[3]{4}) (4 - (2 \sqrt[3]{4}) + \sqrt[3]{16})$

Multiply $2$ by $- 1$ .

$(2 + \sqrt[3]{4}) (4 - 2 \sqrt[3]{4} + \sqrt[3]{16})$

Rewrite $16$ as $2^{3} \cdot 2$ .

Factor $8$ out of $16$ .

$(2 + \sqrt[3]{4}) (4 - 2 \sqrt[3]{4} + \sqrt[3]{8 (2)})$

Rewrite $8$ as $2^{3}$ .

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

Pull terms out from under the radical.

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

Step 2

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

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

Step 3

Simplify terms.

Simplify each term.

Multiply $2$ by $4$ .

$8 + 2 (- 2 \sqrt[3]{4}) + 2 (2 \sqrt[3]{2}) + \sqrt[3]{4} \cdot 4 + \sqrt[3]{4} (- 2 \sqrt[3]{4}) + \sqrt[3]{4} (2 \sqrt[3]{2})$

Multiply $- 2$ by $2$ .

$8 - 4 \sqrt[3]{4} + 2 (2 \sqrt[3]{2}) + \sqrt[3]{4} \cdot 4 + \sqrt[3]{4} (- 2 \sqrt[3]{4}) + \sqrt[3]{4} (2 \sqrt[3]{2})$

Multiply $2$ by $2$ .

$8 - 4 \sqrt[3]{4} + 4 \sqrt[3]{2} + \sqrt[3]{4} \cdot 4 + \sqrt[3]{4} (- 2 \sqrt[3]{4}) + \sqrt[3]{4} (2 \sqrt[3]{2})$

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

$8 - 4 \sqrt[3]{4} + 4 \sqrt[3]{2} + 4 \cdot \sqrt[3]{4} + \sqrt[3]{4} (- 2 \sqrt[3]{4}) + \sqrt[3]{4} (2 \sqrt[3]{2})$

Multiply $\sqrt[3]{4} (- 2 \sqrt[3]{4})$ .

Raise $\sqrt[3]{4}$ to the power of $1$ .

$8 - 4 \sqrt[3]{4} + 4 \sqrt[3]{2} + 4 \sqrt[3]{4} - 2 ({\sqrt[3]{4}}^{1} \sqrt[3]{4}) + \sqrt[3]{4} (2 \sqrt[3]{2})$

Raise $\sqrt[3]{4}$ to the power of $1$ .

$8 - 4 \sqrt[3]{4} + 4 \sqrt[3]{2} + 4 \sqrt[3]{4} - 2 ({\sqrt[3]{4}}^{1} {\sqrt[3]{4}}^{1}) + \sqrt[3]{4} (2 \sqrt[3]{2})$

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

$8 - 4 \sqrt[3]{4} + 4 \sqrt[3]{2} + 4 \sqrt[3]{4} - 2 {\sqrt[3]{4}}^{1 + 1} + \sqrt[3]{4} (2 \sqrt[3]{2})$

Add $1$ and $1$ .

$8 - 4 \sqrt[3]{4} + 4 \sqrt[3]{2} + 4 \sqrt[3]{4} - 2 {\sqrt[3]{4}}^{2} + \sqrt[3]{4} (2 \sqrt[3]{2})$

Rewrite ${\sqrt[3]{4}}^{2}$ as ${(4^{2})}^{\frac{1}{3}}$ .

$8 - 4 \sqrt[3]{4} + 4 \sqrt[3]{2} + 4 \sqrt[3]{4} - 2 \sqrt[3]{4^{2}} + \sqrt[3]{4} (2 \sqrt[3]{2})$

Raise $4$ to the power of $2$ .

$8 - 4 \sqrt[3]{4} + 4 \sqrt[3]{2} + 4 \sqrt[3]{4} - 2 \sqrt[3]{16} + \sqrt[3]{4} (2 \sqrt[3]{2})$

Rewrite $16$ as $2^{3} \cdot 2$ .

Factor $8$ out of $16$ .

$8 - 4 \sqrt[3]{4} + 4 \sqrt[3]{2} + 4 \sqrt[3]{4} - 2 \sqrt[3]{8 (2)} + \sqrt[3]{4} (2 \sqrt[3]{2})$

Rewrite $8$ as $2^{3}$ .

$8 - 4 \sqrt[3]{4} + 4 \sqrt[3]{2} + 4 \sqrt[3]{4} - 2 \sqrt[3]{2^{3} \cdot 2} + \sqrt[3]{4} (2 \sqrt[3]{2})$

Pull terms out from under the radical.

$8 - 4 \sqrt[3]{4} + 4 \sqrt[3]{2} + 4 \sqrt[3]{4} - 2 (2 \sqrt[3]{2}) + \sqrt[3]{4} (2 \sqrt[3]{2})$

Multiply $2$ by $- 2$ .

$8 - 4 \sqrt[3]{4} + 4 \sqrt[3]{2} + 4 \sqrt[3]{4} - 4 \sqrt[3]{2} + \sqrt[3]{4} (2 \sqrt[3]{2})$

Multiply $\sqrt[3]{4} (2 \sqrt[3]{2})$ .

Combine using the product rule for radicals.

$8 - 4 \sqrt[3]{4} + 4 \sqrt[3]{2} + 4 \sqrt[3]{4} - 4 \sqrt[3]{2} + 2 \sqrt[3]{4 \cdot 2}$

Multiply $4$ by $2$ .

$8 - 4 \sqrt[3]{4} + 4 \sqrt[3]{2} + 4 \sqrt[3]{4} - 4 \sqrt[3]{2} + 2 \sqrt[3]{8}$

Rewrite $8$ as $2^{3}$ .

$8 - 4 \sqrt[3]{4} + 4 \sqrt[3]{2} + 4 \sqrt[3]{4} - 4 \sqrt[3]{2} + 2 \sqrt[3]{2^{3}}$

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

$8 - 4 \sqrt[3]{4} + 4 \sqrt[3]{2} + 4 \sqrt[3]{4} - 4 \sqrt[3]{2} + 2 \cdot 2$

Multiply $2$ by $2$ .

$8 - 4 \sqrt[3]{4} + 4 \sqrt[3]{2} + 4 \sqrt[3]{4} - 4 \sqrt[3]{2} + 4$

Simplify by adding terms.

Add $8$ and $4$ .

$12 - 4 \sqrt[3]{4} + 4 \sqrt[3]{2} + 4 \sqrt[3]{4} - 4 \sqrt[3]{2}$

Add $- 4 \sqrt[3]{4}$ and $4 \sqrt[3]{4}$ .

$12 + 0 + 4 \sqrt[3]{2} - 4 \sqrt[3]{2}$

Add $12$ and $0$ .

$12 + 4 \sqrt[3]{2} - 4 \sqrt[3]{2}$

Subtract $4 \sqrt[3]{2}$ from $4 \sqrt[3]{2}$ .

$12 + 0$

Add $12$ and $0$ .

$12$

Simplify ( cube root of 8+ cube root of 4)( cube root of 64- cube root of 32+ cube root of 16)

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