Trigonometry Examples

${(\sin (\frac{π}{3}) \cos (\frac{π}{4}) - \sin (\frac{π}{4}) \cos (\frac{π}{3}))}^{2}$

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

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

${(\frac{\sqrt{3}}{2} \cos (\frac{π}{4}) - \sin (\frac{π}{4}) \cos (\frac{π}{3}))}^{2}$

Step 1.1.2

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

${(\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \sin (\frac{π}{4}) \cos (\frac{π}{3}))}^{2}$

Step 1.1.3

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

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Step 1.1.3.1

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

${(\frac{\sqrt{3} \sqrt{2}}{2 \cdot 2} - \sin (\frac{π}{4}) \cos (\frac{π}{3}))}^{2}$

Step 1.1.3.2

Combine using the product rule for radicals.

${(\frac{\sqrt{3 \cdot 2}}{2 \cdot 2} - \sin (\frac{π}{4}) \cos (\frac{π}{3}))}^{2}$

Step 1.1.3.3

Multiply $3$ by $2$ .

${(\frac{\sqrt{6}}{2 \cdot 2} - \sin (\frac{π}{4}) \cos (\frac{π}{3}))}^{2}$

Step 1.1.3.4

Multiply $2$ by $2$ .

${(\frac{\sqrt{6}}{4} - \sin (\frac{π}{4}) \cos (\frac{π}{3}))}^{2}$

Step 1.1.4

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

${(\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2} \cos (\frac{π}{3}))}^{2}$

Step 1.1.5

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

${(\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})}^{2}$

Step 1.1.6

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

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Step 1.1.6.1

Multiply $\frac{1}{2}$ by $\frac{\sqrt{2}}{2}$ .

${(\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2})}^{2}$

Step 1.1.6.2

Multiply $2$ by $2$ .

${(\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4})}^{2}$

Step 1.2

Rewrite ${(\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4})}^{2}$ as $(\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}) (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4})$ .

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

Step 2

Expand $(\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}) (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $\frac{\sqrt{6}}{4} \cdot \frac{\sqrt{6}}{4}$ .

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Step 3.1.1.1

Multiply $\frac{\sqrt{6}}{4}$ by $\frac{\sqrt{6}}{4}$ .

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

Step 3.1.1.2

Raise $\sqrt{6}$ to the power of $1$ .

$\frac{{\sqrt{6}}^{1} \sqrt{6}}{4 \cdot 4} + \frac{\sqrt{6}}{4} (- \frac{\sqrt{2}}{4}) - \frac{\sqrt{2}}{4} \cdot \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} (- \frac{\sqrt{2}}{4})$

Step 3.1.1.3

Raise $\sqrt{6}$ to the power of $1$ .

$\frac{{\sqrt{6}}^{1} {\sqrt{6}}^{1}}{4 \cdot 4} + \frac{\sqrt{6}}{4} (- \frac{\sqrt{2}}{4}) - \frac{\sqrt{2}}{4} \cdot \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} (- \frac{\sqrt{2}}{4})$

Step 3.1.1.4

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

$\frac{{\sqrt{6}}^{1 + 1}}{4 \cdot 4} + \frac{\sqrt{6}}{4} (- \frac{\sqrt{2}}{4}) - \frac{\sqrt{2}}{4} \cdot \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} (- \frac{\sqrt{2}}{4})$

Step 3.1.1.5

Add $1$ and $1$ .

$\frac{{\sqrt{6}}^{2}}{4 \cdot 4} + \frac{\sqrt{6}}{4} (- \frac{\sqrt{2}}{4}) - \frac{\sqrt{2}}{4} \cdot \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} (- \frac{\sqrt{2}}{4})$

Step 3.1.1.6

Multiply $4$ by $4$ .

$\frac{{\sqrt{6}}^{2}}{16} + \frac{\sqrt{6}}{4} (- \frac{\sqrt{2}}{4}) - \frac{\sqrt{2}}{4} \cdot \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} (- \frac{\sqrt{2}}{4})$

Step 3.1.2

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 3.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\frac{{(6^{\frac{1}{2}})}^{2}}{16} + \frac{\sqrt{6}}{4} (- \frac{\sqrt{2}}{4}) - \frac{\sqrt{2}}{4} \cdot \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} (- \frac{\sqrt{2}}{4})$

Step 3.1.2.2

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

$\frac{6^{\frac{1}{2} \cdot 2}}{16} + \frac{\sqrt{6}}{4} (- \frac{\sqrt{2}}{4}) - \frac{\sqrt{2}}{4} \cdot \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} (- \frac{\sqrt{2}}{4})$

Step 3.1.2.3

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

$\frac{6^{\frac{2}{2}}}{16} + \frac{\sqrt{6}}{4} (- \frac{\sqrt{2}}{4}) - \frac{\sqrt{2}}{4} \cdot \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} (- \frac{\sqrt{2}}{4})$

Step 3.1.2.4

Cancel the common factor of $2$ .

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Step 3.1.2.4.1

Cancel the common factor.

$\frac{6^{\frac{2}{2}}}{16} + \frac{\sqrt{6}}{4} (- \frac{\sqrt{2}}{4}) - \frac{\sqrt{2}}{4} \cdot \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} (- \frac{\sqrt{2}}{4})$

Step 3.1.2.4.2

Rewrite the expression.

$\frac{6^{1}}{16} + \frac{\sqrt{6}}{4} (- \frac{\sqrt{2}}{4}) - \frac{\sqrt{2}}{4} \cdot \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} (- \frac{\sqrt{2}}{4})$

Step 3.1.2.5

Evaluate the exponent.

$\frac{6}{16} + \frac{\sqrt{6}}{4} (- \frac{\sqrt{2}}{4}) - \frac{\sqrt{2}}{4} \cdot \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} (- \frac{\sqrt{2}}{4})$

Step 3.1.3

Cancel the common factor of $6$ and $16$ .

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Step 3.1.3.1

Factor $2$ out of $6$ .

$\frac{2 (3)}{16} + \frac{\sqrt{6}}{4} (- \frac{\sqrt{2}}{4}) - \frac{\sqrt{2}}{4} \cdot \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} (- \frac{\sqrt{2}}{4})$

Step 3.1.3.2

Cancel the common factors.

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Step 3.1.3.2.1

Factor $2$ out of $16$ .

$\frac{2 \cdot 3}{2 \cdot 8} + \frac{\sqrt{6}}{4} (- \frac{\sqrt{2}}{4}) - \frac{\sqrt{2}}{4} \cdot \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} (- \frac{\sqrt{2}}{4})$

Step 3.1.3.2.2

Cancel the common factor.

$\frac{2 \cdot 3}{2 \cdot 8} + \frac{\sqrt{6}}{4} (- \frac{\sqrt{2}}{4}) - \frac{\sqrt{2}}{4} \cdot \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} (- \frac{\sqrt{2}}{4})$

Step 3.1.3.2.3

Rewrite the expression.

$\frac{3}{8} + \frac{\sqrt{6}}{4} (- \frac{\sqrt{2}}{4}) - \frac{\sqrt{2}}{4} \cdot \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} (- \frac{\sqrt{2}}{4})$

Step 3.1.4

Multiply $\frac{\sqrt{6}}{4} (- \frac{\sqrt{2}}{4})$ .

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Step 3.1.4.1

Multiply $\frac{\sqrt{6}}{4}$ by $\frac{\sqrt{2}}{4}$ .

$\frac{3}{8} - \frac{\sqrt{6} \sqrt{2}}{4 \cdot 4} - \frac{\sqrt{2}}{4} \cdot \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} (- \frac{\sqrt{2}}{4})$

Step 3.1.4.2

Combine using the product rule for radicals.

$\frac{3}{8} - \frac{\sqrt{6 \cdot 2}}{4 \cdot 4} - \frac{\sqrt{2}}{4} \cdot \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} (- \frac{\sqrt{2}}{4})$

Step 3.1.4.3

Multiply $6$ by $2$ .

$\frac{3}{8} - \frac{\sqrt{12}}{4 \cdot 4} - \frac{\sqrt{2}}{4} \cdot \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} (- \frac{\sqrt{2}}{4})$

Step 3.1.4.4

Multiply $4$ by $4$ .

$\frac{3}{8} - \frac{\sqrt{12}}{16} - \frac{\sqrt{2}}{4} \cdot \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} (- \frac{\sqrt{2}}{4})$

Step 3.1.5

Simplify the numerator.

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Step 3.1.5.1

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 3.1.5.1.1

Factor $4$ out of $12$ .

$\frac{3}{8} - \frac{\sqrt{4 (3)}}{16} - \frac{\sqrt{2}}{4} \cdot \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} (- \frac{\sqrt{2}}{4})$

Step 3.1.5.1.2

Rewrite $4$ as $2^{2}$ .

$\frac{3}{8} - \frac{\sqrt{2^{2} \cdot 3}}{16} - \frac{\sqrt{2}}{4} \cdot \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} (- \frac{\sqrt{2}}{4})$

Step 3.1.5.2

Pull terms out from under the radical.

$\frac{3}{8} - \frac{2 \sqrt{3}}{16} - \frac{\sqrt{2}}{4} \cdot \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} (- \frac{\sqrt{2}}{4})$

Step 3.1.6

Cancel the common factor of $2$ and $16$ .

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Step 3.1.6.1

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

$\frac{3}{8} - \frac{2 (\sqrt{3})}{16} - \frac{\sqrt{2}}{4} \cdot \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} (- \frac{\sqrt{2}}{4})$

Step 3.1.6.2

Cancel the common factors.

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Step 3.1.6.2.1

Factor $2$ out of $16$ .

$\frac{3}{8} - \frac{2 \sqrt{3}}{2 \cdot 8} - \frac{\sqrt{2}}{4} \cdot \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} (- \frac{\sqrt{2}}{4})$

Step 3.1.6.2.2

Cancel the common factor.

$\frac{3}{8} - \frac{2 \sqrt{3}}{2 \cdot 8} - \frac{\sqrt{2}}{4} \cdot \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} (- \frac{\sqrt{2}}{4})$

Step 3.1.6.2.3

Rewrite the expression.

$\frac{3}{8} - \frac{\sqrt{3}}{8} - \frac{\sqrt{2}}{4} \cdot \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} (- \frac{\sqrt{2}}{4})$

Step 3.1.7

Multiply $- \frac{\sqrt{2}}{4} \cdot \frac{\sqrt{6}}{4}$ .

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Step 3.1.7.1

Multiply $\frac{\sqrt{6}}{4}$ by $\frac{\sqrt{2}}{4}$ .

$\frac{3}{8} - \frac{\sqrt{3}}{8} - \frac{\sqrt{6} \sqrt{2}}{4 \cdot 4} - \frac{\sqrt{2}}{4} (- \frac{\sqrt{2}}{4})$

Step 3.1.7.2

Combine using the product rule for radicals.

$\frac{3}{8} - \frac{\sqrt{3}}{8} - \frac{\sqrt{6 \cdot 2}}{4 \cdot 4} - \frac{\sqrt{2}}{4} (- \frac{\sqrt{2}}{4})$

Step 3.1.7.3

Multiply $6$ by $2$ .

$\frac{3}{8} - \frac{\sqrt{3}}{8} - \frac{\sqrt{12}}{4 \cdot 4} - \frac{\sqrt{2}}{4} (- \frac{\sqrt{2}}{4})$

Step 3.1.7.4

Multiply $4$ by $4$ .

$\frac{3}{8} - \frac{\sqrt{3}}{8} - \frac{\sqrt{12}}{16} - \frac{\sqrt{2}}{4} (- \frac{\sqrt{2}}{4})$

Step 3.1.8

Simplify the numerator.

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Step 3.1.8.1

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 3.1.8.1.1

Factor $4$ out of $12$ .

$\frac{3}{8} - \frac{\sqrt{3}}{8} - \frac{\sqrt{4 (3)}}{16} - \frac{\sqrt{2}}{4} (- \frac{\sqrt{2}}{4})$

Step 3.1.8.1.2

Rewrite $4$ as $2^{2}$ .

$\frac{3}{8} - \frac{\sqrt{3}}{8} - \frac{\sqrt{2^{2} \cdot 3}}{16} - \frac{\sqrt{2}}{4} (- \frac{\sqrt{2}}{4})$

Step 3.1.8.2

Pull terms out from under the radical.

$\frac{3}{8} - \frac{\sqrt{3}}{8} - \frac{2 \sqrt{3}}{16} - \frac{\sqrt{2}}{4} (- \frac{\sqrt{2}}{4})$

Step 3.1.9

Cancel the common factor of $2$ and $16$ .

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Step 3.1.9.1

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

$\frac{3}{8} - \frac{\sqrt{3}}{8} - \frac{2 (\sqrt{3})}{16} - \frac{\sqrt{2}}{4} (- \frac{\sqrt{2}}{4})$

Step 3.1.9.2

Cancel the common factors.

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Step 3.1.9.2.1

Factor $2$ out of $16$ .

$\frac{3}{8} - \frac{\sqrt{3}}{8} - \frac{2 \sqrt{3}}{2 \cdot 8} - \frac{\sqrt{2}}{4} (- \frac{\sqrt{2}}{4})$

Step 3.1.9.2.2

Cancel the common factor.

$\frac{3}{8} - \frac{\sqrt{3}}{8} - \frac{2 \sqrt{3}}{2 \cdot 8} - \frac{\sqrt{2}}{4} (- \frac{\sqrt{2}}{4})$

Step 3.1.9.2.3

Rewrite the expression.

$\frac{3}{8} - \frac{\sqrt{3}}{8} - \frac{\sqrt{3}}{8} - \frac{\sqrt{2}}{4} (- \frac{\sqrt{2}}{4})$

Step 3.1.10

Multiply $- \frac{\sqrt{2}}{4} (- \frac{\sqrt{2}}{4})$ .

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Step 3.1.10.1

Multiply $- 1$ by $- 1$ .

$\frac{3}{8} - \frac{\sqrt{3}}{8} - \frac{\sqrt{3}}{8} + 1 \frac{\sqrt{2}}{4} \frac{\sqrt{2}}{4}$

Step 3.1.10.2

Multiply $\frac{\sqrt{2}}{4}$ by $1$ .

$\frac{3}{8} - \frac{\sqrt{3}}{8} - \frac{\sqrt{3}}{8} + \frac{\sqrt{2}}{4} \cdot \frac{\sqrt{2}}{4}$

Step 3.1.10.3

Multiply $\frac{\sqrt{2}}{4}$ by $\frac{\sqrt{2}}{4}$ .

$\frac{3}{8} - \frac{\sqrt{3}}{8} - \frac{\sqrt{3}}{8} + \frac{\sqrt{2} \sqrt{2}}{4 \cdot 4}$

Step 3.1.10.4

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

$\frac{3}{8} - \frac{\sqrt{3}}{8} - \frac{\sqrt{3}}{8} + \frac{{\sqrt{2}}^{1} \sqrt{2}}{4 \cdot 4}$

Step 3.1.10.5

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

$\frac{3}{8} - \frac{\sqrt{3}}{8} - \frac{\sqrt{3}}{8} + \frac{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}{4 \cdot 4}$

Step 3.1.10.6

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

$\frac{3}{8} - \frac{\sqrt{3}}{8} - \frac{\sqrt{3}}{8} + \frac{{\sqrt{2}}^{1 + 1}}{4 \cdot 4}$

Step 3.1.10.7

Add $1$ and $1$ .

$\frac{3}{8} - \frac{\sqrt{3}}{8} - \frac{\sqrt{3}}{8} + \frac{{\sqrt{2}}^{2}}{4 \cdot 4}$

Step 3.1.10.8

Multiply $4$ by $4$ .

$\frac{3}{8} - \frac{\sqrt{3}}{8} - \frac{\sqrt{3}}{8} + \frac{{\sqrt{2}}^{2}}{16}$

Step 3.1.11

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

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Step 3.1.11.1

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

$\frac{3}{8} - \frac{\sqrt{3}}{8} - \frac{\sqrt{3}}{8} + \frac{{(2^{\frac{1}{2}})}^{2}}{16}$

Step 3.1.11.2

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

$\frac{3}{8} - \frac{\sqrt{3}}{8} - \frac{\sqrt{3}}{8} + \frac{2^{\frac{1}{2} \cdot 2}}{16}$

Step 3.1.11.3

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

$\frac{3}{8} - \frac{\sqrt{3}}{8} - \frac{\sqrt{3}}{8} + \frac{2^{\frac{2}{2}}}{16}$

Step 3.1.11.4

Cancel the common factor of $2$ .

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Step 3.1.11.4.1

Cancel the common factor.

$\frac{3}{8} - \frac{\sqrt{3}}{8} - \frac{\sqrt{3}}{8} + \frac{2^{\frac{2}{2}}}{16}$

Step 3.1.11.4.2

Rewrite the expression.

$\frac{3}{8} - \frac{\sqrt{3}}{8} - \frac{\sqrt{3}}{8} + \frac{2^{1}}{16}$

Step 3.1.11.5

Evaluate the exponent.

$\frac{3}{8} - \frac{\sqrt{3}}{8} - \frac{\sqrt{3}}{8} + \frac{2}{16}$

Step 3.1.12

Cancel the common factor of $2$ and $16$ .

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Step 3.1.12.1

Factor $2$ out of $2$ .

$\frac{3}{8} - \frac{\sqrt{3}}{8} - \frac{\sqrt{3}}{8} + \frac{2 (1)}{16}$

Step 3.1.12.2

Cancel the common factors.

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Step 3.1.12.2.1

Factor $2$ out of $16$ .

$\frac{3}{8} - \frac{\sqrt{3}}{8} - \frac{\sqrt{3}}{8} + \frac{2 \cdot 1}{2 \cdot 8}$

Step 3.1.12.2.2

Cancel the common factor.

$\frac{3}{8} - \frac{\sqrt{3}}{8} - \frac{\sqrt{3}}{8} + \frac{2 \cdot 1}{2 \cdot 8}$

Step 3.1.12.2.3

Rewrite the expression.

$\frac{3}{8} - \frac{\sqrt{3}}{8} - \frac{\sqrt{3}}{8} + \frac{1}{8}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{3 + 1}{8} - \frac{\sqrt{3}}{8} - \frac{\sqrt{3}}{8}$

Step 3.3

Add $3$ and $1$ .

$\frac{4}{8} - \frac{\sqrt{3}}{8} - \frac{\sqrt{3}}{8}$

Step 3.4

Subtract $\frac{\sqrt{3}}{8}$ from $- \frac{\sqrt{3}}{8}$ .

$\frac{4}{8} - 2 \frac{\sqrt{3}}{8}$

Step 4

Simplify each term.

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Step 4.1

Cancel the common factor of $4$ and $8$ .

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Step 4.1.1

Factor $4$ out of $4$ .

$\frac{4 (1)}{8} - 2 \frac{\sqrt{3}}{8}$

Step 4.1.2

Cancel the common factors.

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Step 4.1.2.1

Factor $4$ out of $8$ .

$\frac{4 \cdot 1}{4 \cdot 2} - 2 \frac{\sqrt{3}}{8}$

Step 4.1.2.2

Cancel the common factor.

$\frac{4 \cdot 1}{4 \cdot 2} - 2 \frac{\sqrt{3}}{8}$

Step 4.1.2.3

Rewrite the expression.

$\frac{1}{2} - 2 \frac{\sqrt{3}}{8}$

Step 4.2

Cancel the common factor of $2$ .

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Step 4.2.1

Factor $2$ out of $- 2$ .

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

Step 4.2.2

Factor $2$ out of $8$ .

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

Step 4.2.3

Cancel the common factor.

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

Step 4.2.4

Rewrite the expression.

$\frac{1}{2} - 1 \frac{\sqrt{3}}{4}$

Step 4.3

Rewrite $- 1 \frac{\sqrt{3}}{4}$ as $- \frac{\sqrt{3}}{4}$ .

$\frac{1}{2} - \frac{\sqrt{3}}{4}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2} - \frac{\sqrt{3}}{4}$

Decimal Form:

$0.06698729 \dots$