Algebra Examples

${(\sin (\frac{π}{3}) \tan (\frac{π}{6}) + \csc (\frac{π}{4}))}^{2}$

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

${(\frac{\sqrt{3}}{2} \tan (\frac{π}{6}) + \csc (\frac{π}{4}))}^{2}$

Step 1.1.2

The exact value of $\tan (\frac{π}{6})$ is $\frac{\sqrt{3}}{3}$ .

${(\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{3} + \csc (\frac{π}{4}))}^{2}$

Step 1.1.3

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

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Step 1.1.3.1

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

${(\frac{\sqrt{3} \sqrt{3}}{2 \cdot 3} + \csc (\frac{π}{4}))}^{2}$

Step 1.1.3.2

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

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

Step 1.1.3.3

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

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

Step 1.1.3.4

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

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

Step 1.1.3.5

Add $1$ and $1$ .

${(\frac{{\sqrt{3}}^{2}}{2 \cdot 3} + \csc (\frac{π}{4}))}^{2}$

Step 1.1.3.6

Multiply $2$ by $3$ .

${(\frac{{\sqrt{3}}^{2}}{6} + \csc (\frac{π}{4}))}^{2}$

Step 1.1.4

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

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Step 1.1.4.1

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

${(\frac{{(3^{\frac{1}{2}})}^{2}}{6} + \csc (\frac{π}{4}))}^{2}$

Step 1.1.4.2

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

${(\frac{3^{\frac{1}{2} \cdot 2}}{6} + \csc (\frac{π}{4}))}^{2}$

Step 1.1.4.3

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

${(\frac{3^{\frac{2}{2}}}{6} + \csc (\frac{π}{4}))}^{2}$

Step 1.1.4.4

Cancel the common factor of $2$ .

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Step 1.1.4.4.1

Cancel the common factor.

${(\frac{3^{\frac{2}{2}}}{6} + \csc (\frac{π}{4}))}^{2}$

Step 1.1.4.4.2

Rewrite the expression.

${(\frac{3^{1}}{6} + \csc (\frac{π}{4}))}^{2}$

Step 1.1.4.5

Evaluate the exponent.

${(\frac{3}{6} + \csc (\frac{π}{4}))}^{2}$

Step 1.1.5

Cancel the common factor of $3$ and $6$ .

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Step 1.1.5.1

Factor $3$ out of $3$ .

${(\frac{3 (1)}{6} + \csc (\frac{π}{4}))}^{2}$

Step 1.1.5.2

Cancel the common factors.

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Step 1.1.5.2.1

Factor $3$ out of $6$ .

${(\frac{3 \cdot 1}{3 \cdot 2} + \csc (\frac{π}{4}))}^{2}$

Step 1.1.5.2.2

Cancel the common factor.

${(\frac{3 \cdot 1}{3 \cdot 2} + \csc (\frac{π}{4}))}^{2}$

Step 1.1.5.2.3

Rewrite the expression.

${(\frac{1}{2} + \csc (\frac{π}{4}))}^{2}$

Step 1.1.6

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

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

Step 1.2

Rewrite ${(\frac{1}{2} + \sqrt{2})}^{2}$ as $(\frac{1}{2} + \sqrt{2}) (\frac{1}{2} + \sqrt{2})$ .

$(\frac{1}{2} + \sqrt{2}) (\frac{1}{2} + \sqrt{2})$

Step 2

Expand $(\frac{1}{2} + \sqrt{2}) (\frac{1}{2} + \sqrt{2})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\frac{1}{2} (\frac{1}{2} + \sqrt{2}) + \sqrt{2} (\frac{1}{2} + \sqrt{2})$

Step 2.2

Apply the distributive property.

$\frac{1}{2} \cdot \frac{1}{2} + \frac{1}{2} \sqrt{2} + \sqrt{2} (\frac{1}{2} + \sqrt{2})$

Step 2.3

Apply the distributive property.

$\frac{1}{2} \cdot \frac{1}{2} + \frac{1}{2} \sqrt{2} + \sqrt{2} \frac{1}{2} + \sqrt{2} \sqrt{2}$

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 $\frac{1}{2} \cdot \frac{1}{2}$ .

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Step 3.1.1.1

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

$\frac{1}{2 \cdot 2} + \frac{1}{2} \sqrt{2} + \sqrt{2} \frac{1}{2} + \sqrt{2} \sqrt{2}$

Step 3.1.1.2

Multiply $2$ by $2$ .

$\frac{1}{4} + \frac{1}{2} \sqrt{2} + \sqrt{2} \frac{1}{2} + \sqrt{2} \sqrt{2}$

Step 3.1.2

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

$\frac{1}{4} + \frac{\sqrt{2}}{2} + \sqrt{2} \frac{1}{2} + \sqrt{2} \sqrt{2}$

Step 3.1.3

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

$\frac{1}{4} + \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} + \sqrt{2} \sqrt{2}$

Step 3.1.4

Combine using the product rule for radicals.

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

Step 3.1.5

Multiply $2$ by $2$ .

$\frac{1}{4} + \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} + \sqrt{4}$

Step 3.1.6

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

$\frac{1}{4} + \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} + \sqrt{2^{2}}$

Step 3.1.7

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

$\frac{1}{4} + \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} + 2$

Step 3.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 3.3

Combine $2$ and $\frac{4}{4}$ .

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

Step 3.4

Combine the numerators over the common denominator.

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

Step 3.5

Simplify the numerator.

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Step 3.5.1

Multiply $2$ by $4$ .

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

Step 3.5.2

Add $1$ and $8$ .

$\frac{9}{4} + \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}$

Step 3.6

Combine the numerators over the common denominator.

$\frac{9}{4} + \frac{\sqrt{2} + \sqrt{2}}{2}$

Step 4

Add $\sqrt{2}$ and $\sqrt{2}$ .

$\frac{9}{4} + \frac{2 \sqrt{2}}{2}$

Step 5

Cancel the common factor of $2$ .

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Step 5.1

Cancel the common factor.

$\frac{9}{4} + \frac{2 \sqrt{2}}{2}$

Step 5.2

Divide $\sqrt{2}$ by $1$ .

$\frac{9}{4} + \sqrt{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{9}{4} + \sqrt{2}$

Decimal Form:

$3.66421356 \dots$