Simplify ( cube root of 7- cube root of 10)( cube root of 49+ cube root of 70+ cube root of 100)

$(\sqrt[3]{7} - \sqrt[3]{10}) (\sqrt[3]{49} + \sqrt[3]{70} + \sqrt[3]{100})$

Expand $(\sqrt[3]{7} - \sqrt[3]{10}) (\sqrt[3]{49} + \sqrt[3]{70} + \sqrt[3]{100})$ by multiplying each term in the first expression by each term in the second expression.

$\sqrt[3]{7} \sqrt[3]{49} + \sqrt[3]{7} \sqrt[3]{70} + \sqrt[3]{7} \sqrt[3]{100} - \sqrt[3]{10} \sqrt[3]{49} - \sqrt[3]{10} \sqrt[3]{70} - \sqrt[3]{10} \sqrt[3]{100}$

Simplify terms.

Simplify each term.

Combine using the product rule for radicals.

$\sqrt[3]{7 \cdot 49} + \sqrt[3]{7} \sqrt[3]{70} + \sqrt[3]{7} \sqrt[3]{100} - \sqrt[3]{10} \sqrt[3]{49} - \sqrt[3]{10} \sqrt[3]{70} - \sqrt[3]{10} \sqrt[3]{100}$

Multiply $7$ by $49$ .

$\sqrt[3]{343} + \sqrt[3]{7} \sqrt[3]{70} + \sqrt[3]{7} \sqrt[3]{100} - \sqrt[3]{10} \sqrt[3]{49} - \sqrt[3]{10} \sqrt[3]{70} - \sqrt[3]{10} \sqrt[3]{100}$

Rewrite $343$ as $7^{3}$ .

$\sqrt[3]{7^{3}} + \sqrt[3]{7} \sqrt[3]{70} + \sqrt[3]{7} \sqrt[3]{100} - \sqrt[3]{10} \sqrt[3]{49} - \sqrt[3]{10} \sqrt[3]{70} - \sqrt[3]{10} \sqrt[3]{100}$

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

$7 + \sqrt[3]{7} \sqrt[3]{70} + \sqrt[3]{7} \sqrt[3]{100} - \sqrt[3]{10} \sqrt[3]{49} - \sqrt[3]{10} \sqrt[3]{70} - \sqrt[3]{10} \sqrt[3]{100}$

Combine using the product rule for radicals.

$7 + \sqrt[3]{7 \cdot 70} + \sqrt[3]{7} \sqrt[3]{100} - \sqrt[3]{10} \sqrt[3]{49} - \sqrt[3]{10} \sqrt[3]{70} - \sqrt[3]{10} \sqrt[3]{100}$

Multiply $7$ by $70$ .

$7 + \sqrt[3]{490} + \sqrt[3]{7} \sqrt[3]{100} - \sqrt[3]{10} \sqrt[3]{49} - \sqrt[3]{10} \sqrt[3]{70} - \sqrt[3]{10} \sqrt[3]{100}$

Combine using the product rule for radicals.

$7 + \sqrt[3]{490} + \sqrt[3]{7 \cdot 100} - \sqrt[3]{10} \sqrt[3]{49} - \sqrt[3]{10} \sqrt[3]{70} - \sqrt[3]{10} \sqrt[3]{100}$

Multiply $7$ by $100$ .

$7 + \sqrt[3]{490} + \sqrt[3]{700} - \sqrt[3]{10} \sqrt[3]{49} - \sqrt[3]{10} \sqrt[3]{70} - \sqrt[3]{10} \sqrt[3]{100}$

Multiply $- \sqrt[3]{10} \sqrt[3]{49}$ .

Combine using the product rule for radicals.

$7 + \sqrt[3]{490} + \sqrt[3]{700} - \sqrt[3]{49 \cdot 10} - \sqrt[3]{10} \sqrt[3]{70} - \sqrt[3]{10} \sqrt[3]{100}$

Multiply $49$ by $10$ .

$7 + \sqrt[3]{490} + \sqrt[3]{700} - \sqrt[3]{490} - \sqrt[3]{10} \sqrt[3]{70} - \sqrt[3]{10} \sqrt[3]{100}$

Multiply $- \sqrt[3]{10} \sqrt[3]{70}$ .

Combine using the product rule for radicals.

$7 + \sqrt[3]{490} + \sqrt[3]{700} - \sqrt[3]{490} - \sqrt[3]{70 \cdot 10} - \sqrt[3]{10} \sqrt[3]{100}$

Multiply $70$ by $10$ .

$7 + \sqrt[3]{490} + \sqrt[3]{700} - \sqrt[3]{490} - \sqrt[3]{700} - \sqrt[3]{10} \sqrt[3]{100}$

Multiply $- \sqrt[3]{10} \sqrt[3]{100}$ .

Combine using the product rule for radicals.

$7 + \sqrt[3]{490} + \sqrt[3]{700} - \sqrt[3]{490} - \sqrt[3]{700} - \sqrt[3]{100 \cdot 10}$

Multiply $100$ by $10$ .

$7 + \sqrt[3]{490} + \sqrt[3]{700} - \sqrt[3]{490} - \sqrt[3]{700} - \sqrt[3]{1000}$

Rewrite $1000$ as $10^{3}$ .

$7 + \sqrt[3]{490} + \sqrt[3]{700} - \sqrt[3]{490} - \sqrt[3]{700} - \sqrt[3]{10^{3}}$

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

$7 + \sqrt[3]{490} + \sqrt[3]{700} - \sqrt[3]{490} - \sqrt[3]{700} - 1 \cdot 10$

Multiply $- 1$ by $10$ .

$7 + \sqrt[3]{490} + \sqrt[3]{700} - \sqrt[3]{490} - \sqrt[3]{700} - 10$

Simplify by adding terms.

Subtract $10$ from $7$ .

$- 3 + \sqrt[3]{490} + \sqrt[3]{700} - \sqrt[3]{490} - \sqrt[3]{700}$

Subtract $\sqrt[3]{490}$ from $\sqrt[3]{490}$ .

$- 3 + 0 + \sqrt[3]{700} - \sqrt[3]{700}$

Add $- 3$ and $0$ .

$- 3 + \sqrt[3]{700} - \sqrt[3]{700}$

Subtract $\sqrt[3]{700}$ from $\sqrt[3]{700}$ .

$- 3 + 0$

Add $- 3$ and $0$ .

$- 3$

Simplify ( cube root of 7- cube root of 10)( cube root of 49+ cube root of 70+ cube root of 100)

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