Algebra Examples

$\frac{((- 6) (\sin (\frac{π}{3})) (\sin (\frac{π}{3})) + (6) (\cos (\frac{π}{3})) (\cos (\frac{π}{3})))}{((- 6) (\sin (\frac{π}{3})) (\cos (\frac{π}{3})) - (6) \cos (\frac{π}{3}) \sin (\frac{π}{3}))}$

Step 1

Simplify the numerator.

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Step 1.1

The exact value of $\sin (\frac{π}{3})$ is $\frac{\sqrt{3}}{2}$ .

$\frac{- 6 \frac{\sqrt{3}}{2} \sin (\frac{π}{3}) + (6) (\cos (\frac{π}{3})) (\cos (\frac{π}{3}))}{(- 6) (\sin (\frac{π}{3})) (\cos (\frac{π}{3})) - (6) \cos (\frac{π}{3}) \sin (\frac{π}{3})}$

Step 1.2

Cancel the common factor of $2$ .

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Step 1.2.1

Factor $2$ out of $- 6$ .

$\frac{2 (- 3) \frac{\sqrt{3}}{2} \sin (\frac{π}{3}) + (6) (\cos (\frac{π}{3})) (\cos (\frac{π}{3}))}{(- 6) (\sin (\frac{π}{3})) (\cos (\frac{π}{3})) - (6) \cos (\frac{π}{3}) \sin (\frac{π}{3})}$

Step 1.2.2

Cancel the common factor.

$\frac{2 \cdot - 3 \frac{\sqrt{3}}{2} \sin (\frac{π}{3}) + (6) (\cos (\frac{π}{3})) (\cos (\frac{π}{3}))}{(- 6) (\sin (\frac{π}{3})) (\cos (\frac{π}{3})) - (6) \cos (\frac{π}{3}) \sin (\frac{π}{3})}$

Step 1.2.3

Rewrite the expression.

$\frac{- 3 \sqrt{3} \sin (\frac{π}{3}) + (6) (\cos (\frac{π}{3})) (\cos (\frac{π}{3}))}{(- 6) (\sin (\frac{π}{3})) (\cos (\frac{π}{3})) - (6) \cos (\frac{π}{3}) \sin (\frac{π}{3})}$

Step 1.3

The exact value of $\sin (\frac{π}{3})$ is $\frac{\sqrt{3}}{2}$ .

$\frac{- 3 \sqrt{3} \frac{\sqrt{3}}{2} + (6) (\cos (\frac{π}{3})) (\cos (\frac{π}{3}))}{(- 6) (\sin (\frac{π}{3})) (\cos (\frac{π}{3})) - (6) \cos (\frac{π}{3}) \sin (\frac{π}{3})}$

Step 1.4

Multiply $- 3 \sqrt{3} \frac{\sqrt{3}}{2}$ .

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Step 1.4.1

Combine $\frac{\sqrt{3}}{2}$ and $- 3$ .

$\frac{\frac{\sqrt{3} \cdot - 3}{2} \sqrt{3} + (6) (\cos (\frac{π}{3})) (\cos (\frac{π}{3}))}{(- 6) (\sin (\frac{π}{3})) (\cos (\frac{π}{3})) - (6) \cos (\frac{π}{3}) \sin (\frac{π}{3})}$

Step 1.4.2

Combine $\frac{\sqrt{3} \cdot - 3}{2}$ and $\sqrt{3}$ .

$\frac{\frac{\sqrt{3} \cdot - 3 \sqrt{3}}{2} + (6) (\cos (\frac{π}{3})) (\cos (\frac{π}{3}))}{(- 6) (\sin (\frac{π}{3})) (\cos (\frac{π}{3})) - (6) \cos (\frac{π}{3}) \sin (\frac{π}{3})}$

Step 1.4.3

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

$\frac{\frac{- 3 ({\sqrt{3}}^{1} \sqrt{3})}{2} + (6) (\cos (\frac{π}{3})) (\cos (\frac{π}{3}))}{(- 6) (\sin (\frac{π}{3})) (\cos (\frac{π}{3})) - (6) \cos (\frac{π}{3}) \sin (\frac{π}{3})}$

Step 1.4.4

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

$\frac{\frac{- 3 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}{2} + (6) (\cos (\frac{π}{3})) (\cos (\frac{π}{3}))}{(- 6) (\sin (\frac{π}{3})) (\cos (\frac{π}{3})) - (6) \cos (\frac{π}{3}) \sin (\frac{π}{3})}$

Step 1.4.5

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

$\frac{\frac{- 3 {\sqrt{3}}^{1 + 1}}{2} + (6) (\cos (\frac{π}{3})) (\cos (\frac{π}{3}))}{(- 6) (\sin (\frac{π}{3})) (\cos (\frac{π}{3})) - (6) \cos (\frac{π}{3}) \sin (\frac{π}{3})}$

Step 1.4.6

Add $1$ and $1$ .

$\frac{\frac{- 3 {\sqrt{3}}^{2}}{2} + (6) (\cos (\frac{π}{3})) (\cos (\frac{π}{3}))}{(- 6) (\sin (\frac{π}{3})) (\cos (\frac{π}{3})) - (6) \cos (\frac{π}{3}) \sin (\frac{π}{3})}$

Step 1.5

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

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Step 1.5.1

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

$\frac{\frac{- 3 {(3^{\frac{1}{2}})}^{2}}{2} + (6) (\cos (\frac{π}{3})) (\cos (\frac{π}{3}))}{(- 6) (\sin (\frac{π}{3})) (\cos (\frac{π}{3})) - (6) \cos (\frac{π}{3}) \sin (\frac{π}{3})}$

Step 1.5.2

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

$\frac{\frac{- 3 \cdot 3^{\frac{1}{2} \cdot 2}}{2} + (6) (\cos (\frac{π}{3})) (\cos (\frac{π}{3}))}{(- 6) (\sin (\frac{π}{3})) (\cos (\frac{π}{3})) - (6) \cos (\frac{π}{3}) \sin (\frac{π}{3})}$

Step 1.5.3

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

$\frac{\frac{- 3 \cdot 3^{\frac{2}{2}}}{2} + (6) (\cos (\frac{π}{3})) (\cos (\frac{π}{3}))}{(- 6) (\sin (\frac{π}{3})) (\cos (\frac{π}{3})) - (6) \cos (\frac{π}{3}) \sin (\frac{π}{3})}$

Step 1.5.4

Cancel the common factor of $2$ .

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Step 1.5.4.1

Cancel the common factor.

$\frac{\frac{- 3 \cdot 3^{\frac{2}{2}}}{2} + (6) (\cos (\frac{π}{3})) (\cos (\frac{π}{3}))}{(- 6) (\sin (\frac{π}{3})) (\cos (\frac{π}{3})) - (6) \cos (\frac{π}{3}) \sin (\frac{π}{3})}$

Step 1.5.4.2

Rewrite the expression.

$\frac{\frac{- 3 \cdot 3^{1}}{2} + (6) (\cos (\frac{π}{3})) (\cos (\frac{π}{3}))}{(- 6) (\sin (\frac{π}{3})) (\cos (\frac{π}{3})) - (6) \cos (\frac{π}{3}) \sin (\frac{π}{3})}$

Step 1.5.5

Evaluate the exponent.

$\frac{\frac{- 3 \cdot 3}{2} + (6) (\cos (\frac{π}{3})) (\cos (\frac{π}{3}))}{(- 6) (\sin (\frac{π}{3})) (\cos (\frac{π}{3})) - (6) \cos (\frac{π}{3}) \sin (\frac{π}{3})}$

Step 1.6

Multiply $- 3$ by $3$ .

$\frac{\frac{- 9}{2} + (6) (\cos (\frac{π}{3})) (\cos (\frac{π}{3}))}{(- 6) (\sin (\frac{π}{3})) (\cos (\frac{π}{3})) - (6) \cos (\frac{π}{3}) \sin (\frac{π}{3})}$

Step 1.7

Move the negative in front of the fraction.

$\frac{- \frac{9}{2} + (6) (\cos (\frac{π}{3})) (\cos (\frac{π}{3}))}{(- 6) (\sin (\frac{π}{3})) (\cos (\frac{π}{3})) - (6) \cos (\frac{π}{3}) \sin (\frac{π}{3})}$

Step 1.8

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$\frac{- \frac{9}{2} + 6 (\frac{1}{2}) \cos (\frac{π}{3})}{(- 6) (\sin (\frac{π}{3})) (\cos (\frac{π}{3})) - (6) \cos (\frac{π}{3}) \sin (\frac{π}{3})}$

Step 1.9

Cancel the common factor of $2$ .

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Step 1.9.1

Factor $2$ out of $6$ .

$\frac{- \frac{9}{2} + 2 (3) \frac{1}{2} \cos (\frac{π}{3})}{(- 6) (\sin (\frac{π}{3})) (\cos (\frac{π}{3})) - (6) \cos (\frac{π}{3}) \sin (\frac{π}{3})}$

Step 1.9.2

Cancel the common factor.

$\frac{- \frac{9}{2} + 2 \cdot 3 \frac{1}{2} \cos (\frac{π}{3})}{(- 6) (\sin (\frac{π}{3})) (\cos (\frac{π}{3})) - (6) \cos (\frac{π}{3}) \sin (\frac{π}{3})}$

Step 1.9.3

Rewrite the expression.

$\frac{- \frac{9}{2} + 3 \cos (\frac{π}{3})}{(- 6) (\sin (\frac{π}{3})) (\cos (\frac{π}{3})) - (6) \cos (\frac{π}{3}) \sin (\frac{π}{3})}$

Step 1.10

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$\frac{- \frac{9}{2} + 3 (\frac{1}{2})}{(- 6) (\sin (\frac{π}{3})) (\cos (\frac{π}{3})) - (6) \cos (\frac{π}{3}) \sin (\frac{π}{3})}$

Step 1.11

Combine $3$ and $\frac{1}{2}$ .

$\frac{- \frac{9}{2} + \frac{3}{2}}{(- 6) (\sin (\frac{π}{3})) (\cos (\frac{π}{3})) - (6) \cos (\frac{π}{3}) \sin (\frac{π}{3})}$

Step 1.12

Combine the numerators over the common denominator.

$\frac{\frac{- 9 + 3}{2}}{(- 6) (\sin (\frac{π}{3})) (\cos (\frac{π}{3})) - (6) \cos (\frac{π}{3}) \sin (\frac{π}{3})}$

Step 1.13

Add $- 9$ and $3$ .

$\frac{\frac{- 6}{2}}{(- 6) (\sin (\frac{π}{3})) (\cos (\frac{π}{3})) - (6) \cos (\frac{π}{3}) \sin (\frac{π}{3})}$

Step 1.14

Divide $- 6$ by $2$ .

$\frac{- 3}{(- 6) (\sin (\frac{π}{3})) (\cos (\frac{π}{3})) - (6) \cos (\frac{π}{3}) \sin (\frac{π}{3})}$

Step 2

Simplify the denominator.

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Step 2.1

The exact value of $\sin (\frac{π}{3})$ is $\frac{\sqrt{3}}{2}$ .

$\frac{- 3}{- 6 \frac{\sqrt{3}}{2} \cos (\frac{π}{3}) - (6) \cos (\frac{π}{3}) \sin (\frac{π}{3})}$

Step 2.2

Cancel the common factor of $2$ .

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Step 2.2.1

Factor $2$ out of $- 6$ .

$\frac{- 3}{2 (- 3) \frac{\sqrt{3}}{2} \cos (\frac{π}{3}) - (6) \cos (\frac{π}{3}) \sin (\frac{π}{3})}$

Step 2.2.2

Cancel the common factor.

$\frac{- 3}{2 \cdot - 3 \frac{\sqrt{3}}{2} \cos (\frac{π}{3}) - (6) \cos (\frac{π}{3}) \sin (\frac{π}{3})}$

Step 2.2.3

Rewrite the expression.

$\frac{- 3}{- 3 \sqrt{3} \cos (\frac{π}{3}) - (6) \cos (\frac{π}{3}) \sin (\frac{π}{3})}$

Step 2.3

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$\frac{- 3}{- 3 \sqrt{3} \frac{1}{2} - (6) \cos (\frac{π}{3}) \sin (\frac{π}{3})}$

Step 2.4

Multiply $- 3 \sqrt{3} \frac{1}{2}$ .

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Step 2.4.1

Combine $\frac{1}{2}$ and $- 3$ .

$\frac{- 3}{\frac{- 3}{2} \sqrt{3} - (6) \cos (\frac{π}{3}) \sin (\frac{π}{3})}$

Step 2.4.2

Combine $\frac{- 3}{2}$ and $\sqrt{3}$ .

$\frac{- 3}{\frac{- 3 \sqrt{3}}{2} - (6) \cos (\frac{π}{3}) \sin (\frac{π}{3})}$

Step 2.5

Move the negative in front of the fraction.

$\frac{- 3}{- \frac{3 \sqrt{3}}{2} - (6) \cos (\frac{π}{3}) \sin (\frac{π}{3})}$

Step 2.6

Multiply $- 1$ by $6$ .

$\frac{- 3}{- \frac{3 \sqrt{3}}{2} - 6 \cos (\frac{π}{3}) \sin (\frac{π}{3})}$

Step 2.7

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$\frac{- 3}{- \frac{3 \sqrt{3}}{2} - 6 (\frac{1}{2}) \sin (\frac{π}{3})}$

Step 2.8

Cancel the common factor of $2$ .

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Step 2.8.1

Factor $2$ out of $- 6$ .

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

Step 2.8.2

Cancel the common factor.

$\frac{- 3}{- \frac{3 \sqrt{3}}{2} + 2 \cdot - 3 \frac{1}{2} \sin (\frac{π}{3})}$

Step 2.8.3

Rewrite the expression.

$\frac{- 3}{- \frac{3 \sqrt{3}}{2} - 3 \sin (\frac{π}{3})}$

Step 2.9

The exact value of $\sin (\frac{π}{3})$ is $\frac{\sqrt{3}}{2}$ .

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

Step 2.10

Combine $- 3$ and $\frac{\sqrt{3}}{2}$ .

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

Step 2.11

Move the negative in front of the fraction.

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

Step 2.12

Subtract $\frac{3 \sqrt{3}}{2}$ from $- \frac{3 \sqrt{3}}{2}$ .

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

Step 3

Combine $- 2$ and $\frac{3 \sqrt{3}}{2}$ .

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

Step 4

Multiply $- 2$ by $3$ .

$\frac{- 3}{\frac{- 6 \sqrt{3}}{2}}$

Step 5

Reduce the expression by cancelling the common factors.

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Step 5.1

Reduce the expression $\frac{- 6 \sqrt{3}}{2}$ by cancelling the common factors.

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Step 5.1.1

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

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

Step 5.1.2

Factor $2$ out of $2$ .

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

Step 5.1.3

Cancel the common factor.

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

Step 5.1.4

Rewrite the expression.

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

Step 5.2

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

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

Step 6

Cancel the common factor of $- 3$ .

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

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

Step 7

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

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

Step 8

Combine and simplify the denominator.

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

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

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

Step 8.4

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

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

Step 8.5

Add $1$ and $1$ .

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

Step 8.6

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

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Step 8.6.1

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

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

Step 8.6.2

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

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

Step 8.6.3

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

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

Step 8.6.4

Cancel the common factor of $2$ .

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Step 8.6.4.1

Cancel the common factor.

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

Step 8.6.4.2

Rewrite the expression.

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

Step 8.6.5

Evaluate the exponent.

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

Step 9

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.57735026 \dots$