Algebra Examples

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

Step 1

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.

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

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Combine using the product rule for radicals.

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

Step 2.1.2

Multiply $7$ by $49$ .

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

Step 2.1.3

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

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

Step 2.1.4

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

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

Step 2.1.5

Combine using the product rule for radicals.

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

Step 2.1.6

Multiply $7$ by $70$ .

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

Step 2.1.7

Combine using the product rule for radicals.

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

Step 2.1.8

Multiply $7$ by $100$ .

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

Step 2.1.9

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

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Step 2.1.9.1

Combine using the product rule for radicals.

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

Step 2.1.9.2

Multiply $49$ by $10$ .

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

Step 2.1.10

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

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Step 2.1.10.1

Combine using the product rule for radicals.

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

Step 2.1.10.2

Multiply $70$ by $10$ .

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

Step 2.1.11

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

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Step 2.1.11.1

Combine using the product rule for radicals.

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

Step 2.1.11.2

Multiply $100$ by $10$ .

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

Step 2.1.12

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

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

Step 2.1.13

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

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

Step 2.1.14

Multiply $- 1$ by $10$ .

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

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Subtract $10$ from $7$ .

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

Step 2.2.2

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

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

Step 2.2.3

Add $- 3$ and $0$ .

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

Step 2.2.4

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

$- 3 + 0$

Step 2.2.5

Add $- 3$ and $0$ .

$- 3$