Trigonometry Examples

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

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 1.2

Combine and simplify the denominator.

Tap for more steps...

Step 1.2.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

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

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

Step 1.2.5

Add $1$ and $1$ .

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

Step 1.2.6

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

Tap for more steps...

Step 1.2.6.1

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

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

Step 1.2.6.2

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

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

Step 1.2.6.3

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

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

Step 1.2.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.2.6.4.1

Cancel the common factor.

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

Step 1.2.6.4.2

Rewrite the expression.

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

Step 1.2.6.5

Evaluate the exponent.

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

Step 2

Simplify terms.

Tap for more steps...

Step 2.1

Write $1$ as a fraction with a common denominator.

$\sqrt{3} (\frac{3}{3} - \frac{\sqrt{3}}{3})$

Step 2.2

Combine the numerators over the common denominator.

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

Step 2.3

Combine $\sqrt{3}$ and $\frac{3 - \sqrt{3}}{3}$ .

$\frac{\sqrt{3} (3 - \sqrt{3})}{3}$

Step 2.4

Apply the distributive property.

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

Step 2.5

Move $3$ to the left of $\sqrt{3}$ .

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

Step 3

Multiply $\sqrt{3} (- \sqrt{3})$ .

Tap for more steps...

Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

$\frac{3 \cdot \sqrt{3} - {\sqrt{3}}^{1 + 1}}{3}$

Step 3.4

Add $1$ and $1$ .

$\frac{3 \cdot \sqrt{3} - {\sqrt{3}}^{2}}{3}$

Step 4

Simplify each term.

Tap for more steps...

Step 4.1

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

Tap for more steps...

Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.1.3

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

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

Step 4.1.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.1.4.1

Cancel the common factor.

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

Step 4.1.4.2

Rewrite the expression.

$\frac{3 \sqrt{3} - 3^{1}}{3}$

Step 4.1.5

Evaluate the exponent.

$\frac{3 \sqrt{3} - 1 \cdot 3}{3}$

Step 4.2

Multiply $- 1$ by $3$ .

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

Step 5

Cancel the common factor of $3 \sqrt{3} - 3$ and $3$ .

Tap for more steps...

Step 5.1

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

$\frac{3 (\sqrt{3}) - 3}{3}$

Step 5.2

Factor $3$ out of $- 3$ .

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

Step 5.3

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

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

Step 5.4

Cancel the common factors.

Tap for more steps...

Step 5.4.1

Factor $3$ out of $3$ .

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

Step 5.4.2

Cancel the common factor.

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

Step 5.4.3

Rewrite the expression.

$\frac{\sqrt{3} - 1}{1}$

Step 5.4.4

Divide $\sqrt{3} - 1$ by $1$ .

$\sqrt{3} - 1$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\sqrt{3} - 1$

Decimal Form:

$0.73205080 \dots$