Algebra Examples

${(\cos (\frac{π}{4}) + \sin (\frac{π}{6}))}^{2}$

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 $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

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

Step 1.1.2

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

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

Step 1.2

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

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

Step 2

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

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Step 2.1

Apply the distributive property.

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

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

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

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Step 3.1.1.1

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

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

Step 3.1.1.2

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

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

Step 3.1.1.3

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

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

Step 3.1.1.4

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

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

Step 3.1.1.5

Add $1$ and $1$ .

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

Step 3.1.1.6

Multiply $2$ by $2$ .

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

Step 3.1.2

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

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Step 3.1.2.1

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

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

Step 3.1.2.2

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

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

Step 3.1.2.3

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

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

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.

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

Step 3.1.2.4.2

Rewrite the expression.

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

Step 3.1.2.5

Evaluate the exponent.

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

Step 3.1.3

Cancel the common factor of $2$ and $4$ .

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Step 3.1.3.1

Factor $2$ out of $2$ .

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

Step 3.1.3.2

Cancel the common factors.

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Step 3.1.3.2.1

Factor $2$ out of $4$ .

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

Step 3.1.3.2.2

Cancel the common factor.

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

Step 3.1.3.2.3

Rewrite the expression.

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

Step 3.1.4

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

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Step 3.1.4.1

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

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

Step 3.1.4.2

Multiply $2$ by $2$ .

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

Step 3.1.5

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

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Step 3.1.5.1

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

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

Step 3.1.5.2

Multiply $2$ by $2$ .

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

Step 3.1.6

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

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Step 3.1.6.1

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

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

Step 3.1.6.2

Multiply $2$ by $2$ .

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

Step 3.2

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 3.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.1

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

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

Step 3.3.2

Multiply $2$ by $2$ .

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

Step 3.4

Combine the numerators over the common denominator.

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

Step 3.5

Add $2$ and $1$ .

$\frac{3}{4} + \frac{\sqrt{2}}{4} + \frac{\sqrt{2}}{4}$

Step 3.6

Combine the numerators over the common denominator.

$\frac{3}{4} + \frac{\sqrt{2} + \sqrt{2}}{4}$

Step 4

Combine the numerators over the common denominator.

$\frac{3 + \sqrt{2} + \sqrt{2}}{4}$

Step 5

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

$\frac{3 + 2 \sqrt{2}}{4}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{3 + 2 \sqrt{2}}{4}$

Decimal Form:

$1.45710678 \dots$