Trigonometry Examples

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

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.

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

Step 2

Simplify each term.

Tap for more steps...

Step 2.1

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

Tap for more steps...

Step 2.1.1

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

Tap for more steps...

Step 2.1.1.1

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

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

Step 2.1.1.2

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

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

Step 2.1.2

Add $1$ and $2$ .

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

Step 2.2

Rewrite using the commutative property of multiplication.

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

Step 2.3

Multiply $x$ by $x$ by adding the exponents.

Tap for more steps...

Step 2.3.1

Move $x$ .

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

Step 2.3.2

Multiply $x$ by $x$ .

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

Step 2.4

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

Tap for more steps...

Step 2.4.1

Raise $\sqrt[3]{2}$ to the power of $1$ .

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

Step 2.4.2

Raise $\sqrt[3]{2}$ to the power of $1$ .

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

Step 2.4.3

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

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

Step 2.4.4

Add $1$ and $1$ .

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

Step 2.5

Rewrite ${\sqrt[3]{2}}^{2}$ as $\sqrt[3]{2^{2}}$ .

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

Step 2.6

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

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

Step 2.7

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

Tap for more steps...

Step 2.7.1

Combine using the product rule for radicals.

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

Step 2.7.2

Multiply $4$ by $2$ .

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

Step 2.8

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

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

Step 2.9

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

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

Step 2.10

Multiply $- 1$ by $2$ .

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

Step 3

Combine the opposite terms in $x^{3} + \sqrt[3]{2} x^{2} + x \sqrt[3]{4} - \sqrt[3]{2} x^{2} - \sqrt[3]{4} x - 2$ .

Tap for more steps...

Step 3.1

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

$x^{3} - 2 = x^{3} + x \sqrt[3]{4} + 0 - \sqrt[3]{4} x - 2$

Step 3.2

Add $x^{3} + x \sqrt[3]{4}$ and $0$ .

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

Step 3.3

Reorder the factors in the terms $x \sqrt[3]{4}$ and $- \sqrt[3]{4} x$ .

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

Step 3.4

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

$x^{3} - 2 = x^{3} + 0 - 2$

Step 3.5

Add $x^{3}$ and $0$ .

$x^{3} - 2 = x^{3} - 2$

Step 4

Since the two sides have been shown to be equivalent, the equation is an identity.

$x^{3} - 2 = (x - \sqrt[3]{2}) (x^{2} + \sqrt[3]{2} x + \sqrt[3]{4})$ is an identity.