Trigonometry Examples

$2 {(\frac{\sqrt{3} - 1}{2})}^{3}$

Step 1

Simplify the expression.

Tap for more steps...

Step 1.1

Apply the product rule to $\frac{\sqrt{3} - 1}{2}$ .

$2 \frac{{(\sqrt{3} - 1)}^{3}}{2^{3}}$

Step 1.2

Raise $2$ to the power of $3$ .

$2 \frac{{(\sqrt{3} - 1)}^{3}}{8}$

Step 2

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.1

Factor $2$ out of $8$ .

$2 \frac{{(\sqrt{3} - 1)}^{3}}{2 (4)}$

Step 2.2

Cancel the common factor.

$2 \frac{{(\sqrt{3} - 1)}^{3}}{2 \cdot 4}$

Step 2.3

Rewrite the expression.

$\frac{{(\sqrt{3} - 1)}^{3}}{4}$

Step 3

Simplify ${(\sqrt{3} - 1)}^{3}$ .

Tap for more steps...

Step 3.1

Use the Binomial Theorem.

$\frac{{\sqrt{3}}^{3} + 3 {\sqrt{3}}^{2} \cdot - 1 + 3 \sqrt{3} {(- 1)}^{2} + {(- 1)}^{3}}{4}$

Step 3.2

Simplify each term.

Tap for more steps...

Step 3.2.1

Rewrite ${\sqrt{3}}^{3}$ as $\sqrt{3^{3}}$ .

$\frac{\sqrt{3^{3}} + 3 {\sqrt{3}}^{2} \cdot - 1 + 3 \sqrt{3} {(- 1)}^{2} + {(- 1)}^{3}}{4}$

Step 3.2.2

Raise $3$ to the power of $3$ .

$\frac{\sqrt{27} + 3 {\sqrt{3}}^{2} \cdot - 1 + 3 \sqrt{3} {(- 1)}^{2} + {(- 1)}^{3}}{4}$

Step 3.2.3

Rewrite $27$ as $3^{2} \cdot 3$ .

Tap for more steps...

Step 3.2.3.1

Factor $9$ out of $27$ .

$\frac{\sqrt{9 (3)} + 3 {\sqrt{3}}^{2} \cdot - 1 + 3 \sqrt{3} {(- 1)}^{2} + {(- 1)}^{3}}{4}$

Step 3.2.3.2

Rewrite $9$ as $3^{2}$ .

$\frac{\sqrt{3^{2} \cdot 3} + 3 {\sqrt{3}}^{2} \cdot - 1 + 3 \sqrt{3} {(- 1)}^{2} + {(- 1)}^{3}}{4}$

Step 3.2.4

Pull terms out from under the radical.

$\frac{3 \sqrt{3} + 3 {\sqrt{3}}^{2} \cdot - 1 + 3 \sqrt{3} {(- 1)}^{2} + {(- 1)}^{3}}{4}$

Step 3.2.5

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

Tap for more steps...

Step 3.2.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\frac{3 \sqrt{3} + 3 {(3^{\frac{1}{2}})}^{2} \cdot - 1 + 3 \sqrt{3} {(- 1)}^{2} + {(- 1)}^{3}}{4}$

Step 3.2.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{3 \sqrt{3} + 3 \cdot 3^{\frac{1}{2} \cdot 2} \cdot - 1 + 3 \sqrt{3} {(- 1)}^{2} + {(- 1)}^{3}}{4}$

Step 3.2.5.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{3 \sqrt{3} + 3 \cdot 3^{\frac{2}{2}} \cdot - 1 + 3 \sqrt{3} {(- 1)}^{2} + {(- 1)}^{3}}{4}$

Step 3.2.5.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.5.4.1

Cancel the common factor.

$\frac{3 \sqrt{3} + 3 \cdot 3^{\frac{2}{2}} \cdot - 1 + 3 \sqrt{3} {(- 1)}^{2} + {(- 1)}^{3}}{4}$

Step 3.2.5.4.2

Rewrite the expression.

$\frac{3 \sqrt{3} + 3 \cdot 3^{1} \cdot - 1 + 3 \sqrt{3} {(- 1)}^{2} + {(- 1)}^{3}}{4}$

Step 3.2.5.5

Evaluate the exponent.

$\frac{3 \sqrt{3} + 3 \cdot 3 \cdot - 1 + 3 \sqrt{3} {(- 1)}^{2} + {(- 1)}^{3}}{4}$

Step 3.2.6

Multiply $3 \cdot 3 \cdot - 1$ .

Tap for more steps...

Step 3.2.6.1

Multiply $3$ by $3$ .

$\frac{3 \sqrt{3} + 9 \cdot - 1 + 3 \sqrt{3} {(- 1)}^{2} + {(- 1)}^{3}}{4}$

Step 3.2.6.2

Multiply $9$ by $- 1$ .

$\frac{3 \sqrt{3} - 9 + 3 \sqrt{3} {(- 1)}^{2} + {(- 1)}^{3}}{4}$

Step 3.2.7

Raise $- 1$ to the power of $2$ .

$\frac{3 \sqrt{3} - 9 + 3 \sqrt{3} \cdot 1 + {(- 1)}^{3}}{4}$

Step 3.2.8

Multiply $3$ by $1$ .

$\frac{3 \sqrt{3} - 9 + 3 \sqrt{3} + {(- 1)}^{3}}{4}$

Step 3.2.9

Raise $- 1$ to the power of $3$ .

$\frac{3 \sqrt{3} - 9 + 3 \sqrt{3} - 1}{4}$

Step 3.3

Add $3 \sqrt{3}$ and $3 \sqrt{3}$ .

$\frac{- 9 + 6 \sqrt{3} - 1}{4}$

Step 3.4

Subtract $1$ from $- 9$ .

$\frac{- 10 + 6 \sqrt{3}}{4}$

Step 4

Cancel the common factor of $- 10 + 6 \sqrt{3}$ and $4$ .

Tap for more steps...

Step 4.1

Factor $2$ out of $- 10$ .

$\frac{2 (- 5) + 6 \sqrt{3}}{4}$

Step 4.2

Factor $2$ out of $6 \sqrt{3}$ .

$\frac{2 (- 5) + 2 (3 \sqrt{3})}{4}$

Step 4.3

Factor $2$ out of $2 (- 5) + 2 (3 \sqrt{3})$ .

$\frac{2 (- 5 + 3 \sqrt{3})}{4}$

Step 4.4

Cancel the common factors.

Tap for more steps...

Step 4.4.1

Factor $2$ out of $4$ .

$\frac{2 (- 5 + 3 \sqrt{3})}{2 \cdot 2}$

Step 4.4.2

Cancel the common factor.

$\frac{2 (- 5 + 3 \sqrt{3})}{2 \cdot 2}$

Step 4.4.3

Rewrite the expression.

$\frac{- 5 + 3 \sqrt{3}}{2}$

Step 5

Rewrite $- 5$ as $- 1 (5)$ .

$\frac{- 1 (5) + 3 \sqrt{3}}{2}$

Step 6

Factor $- 1$ out of $3 \sqrt{3}$ .

$\frac{- 1 (5) - (- 3 \sqrt{3})}{2}$

Step 7

Factor $- 1$ out of $- 1 (5) - (- 3 \sqrt{3})$ .

$\frac{- 1 (5 - 3 \sqrt{3})}{2}$

Step 8

Move the negative in front of the fraction.

$- \frac{5 - 3 \sqrt{3}}{2}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- \frac{5 - 3 \sqrt{3}}{2}$

Decimal Form:

$0.09807621 \dots$