Simplify (x- cube root of 2)(x^2+ cube root of 2x+ cube root of 4)

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

Step 1

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

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

Step 2

Simplify terms.

Simplify each term.

Multiply $x$ by $x^{2}$ by adding the exponents.

Multiply $x$ by $x^{2}$ .

Raise $x$ to the power of $1$ .

$x^{1} x^{2} + x \sqrt[3]{2 x} + x \sqrt[3]{4} - \sqrt[3]{2} x^{2} - \sqrt[3]{2} \sqrt[3]{2 x} - \sqrt[3]{2} \sqrt[3]{4}$

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

$x^{1 + 2} + x \sqrt[3]{2 x} + x \sqrt[3]{4} - \sqrt[3]{2} x^{2} - \sqrt[3]{2} \sqrt[3]{2 x} - \sqrt[3]{2} \sqrt[3]{4}$

Add $1$ and $2$ .

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

Multiply $- \sqrt[3]{2} \sqrt[3]{2 x}$ .

Combine using the product rule for radicals.

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

Multiply $2$ by $2$ .

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

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

Combine using the product rule for radicals.

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

Multiply $4$ by $2$ .

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

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

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

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

$x^{3} + x \sqrt[3]{2 x} + x \sqrt[3]{4} - \sqrt[3]{2} x^{2} - \sqrt[3]{4 x} - 1 \cdot 2$

Multiply $- 1$ by $2$ .

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

Reorder.

Move $x \sqrt[3]{2 x}$ .

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

Move $x \sqrt[3]{4}$ .

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

Simplify (x- cube root of 2)(x^2+ cube root of 2x+ cube root of 4)

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