Trigonometry Examples

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

Step 1

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

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

Step 2

Combine fractions.

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Step 2.1

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

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

Step 2.2

Expand the denominator using the FOIL method.

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

Step 2.3

Simplify.

$\frac{(4 \sqrt{2} + 1) (4 \sqrt{2} + 1)}{31}$

Step 3

Simplify the numerator.

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Step 3.1

Raise $4 \sqrt{2} + 1$ to the power of $1$ .

$\frac{{(4 \sqrt{2} + 1)}^{1} (4 \sqrt{2} + 1)}{31}$

Step 3.2

Raise $4 \sqrt{2} + 1$ to the power of $1$ .

$\frac{{(4 \sqrt{2} + 1)}^{1} {(4 \sqrt{2} + 1)}^{1}}{31}$

Step 3.3

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

$\frac{{(4 \sqrt{2} + 1)}^{1 + 1}}{31}$

Step 3.4

Add $1$ and $1$ .

$\frac{{(4 \sqrt{2} + 1)}^{2}}{31}$

Step 4

Simplify ${(4 \sqrt{2} + 1)}^{2}$ .

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Step 4.1

Rewrite ${(4 \sqrt{2} + 1)}^{2}$ as $(4 \sqrt{2} + 1) (4 \sqrt{2} + 1)$ .

$\frac{(4 \sqrt{2} + 1) (4 \sqrt{2} + 1)}{31}$

Step 4.2

Expand $(4 \sqrt{2} + 1) (4 \sqrt{2} + 1)$ using the FOIL Method.

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Step 4.2.1

Apply the distributive property.

$\frac{4 \sqrt{2} (4 \sqrt{2} + 1) + 1 (4 \sqrt{2} + 1)}{31}$

Step 4.2.2

Apply the distributive property.

$\frac{4 \sqrt{2} (4 \sqrt{2}) + 4 \sqrt{2} \cdot 1 + 1 (4 \sqrt{2} + 1)}{31}$

Step 4.2.3

Apply the distributive property.

$\frac{4 \sqrt{2} (4 \sqrt{2}) + 4 \sqrt{2} \cdot 1 + 1 (4 \sqrt{2}) + 1 \cdot 1}{31}$

Step 4.3

Simplify and combine like terms.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

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

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Step 4.3.1.1.1

Multiply $4$ by $4$ .

$\frac{16 \sqrt{2} \sqrt{2} + 4 \sqrt{2} \cdot 1 + 1 (4 \sqrt{2}) + 1 \cdot 1}{31}$

Step 4.3.1.1.2

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

$\frac{16 ({\sqrt{2}}^{1} \sqrt{2}) + 4 \sqrt{2} \cdot 1 + 1 (4 \sqrt{2}) + 1 \cdot 1}{31}$

Step 4.3.1.1.3

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

$\frac{16 ({\sqrt{2}}^{1} {\sqrt{2}}^{1}) + 4 \sqrt{2} \cdot 1 + 1 (4 \sqrt{2}) + 1 \cdot 1}{31}$

Step 4.3.1.1.4

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

$\frac{16 {\sqrt{2}}^{1 + 1} + 4 \sqrt{2} \cdot 1 + 1 (4 \sqrt{2}) + 1 \cdot 1}{31}$

Step 4.3.1.1.5

Add $1$ and $1$ .

$\frac{16 {\sqrt{2}}^{2} + 4 \sqrt{2} \cdot 1 + 1 (4 \sqrt{2}) + 1 \cdot 1}{31}$

Step 4.3.1.2

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

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Step 4.3.1.2.1

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

$\frac{16 {(2^{\frac{1}{2}})}^{2} + 4 \sqrt{2} \cdot 1 + 1 (4 \sqrt{2}) + 1 \cdot 1}{31}$

Step 4.3.1.2.2

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

$\frac{16 \cdot 2^{\frac{1}{2} \cdot 2} + 4 \sqrt{2} \cdot 1 + 1 (4 \sqrt{2}) + 1 \cdot 1}{31}$

Step 4.3.1.2.3

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

$\frac{16 \cdot 2^{\frac{2}{2}} + 4 \sqrt{2} \cdot 1 + 1 (4 \sqrt{2}) + 1 \cdot 1}{31}$

Step 4.3.1.2.4

Cancel the common factor of $2$ .

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Step 4.3.1.2.4.1

Cancel the common factor.

$\frac{16 \cdot 2^{\frac{2}{2}} + 4 \sqrt{2} \cdot 1 + 1 (4 \sqrt{2}) + 1 \cdot 1}{31}$

Step 4.3.1.2.4.2

Rewrite the expression.

$\frac{16 \cdot 2^{1} + 4 \sqrt{2} \cdot 1 + 1 (4 \sqrt{2}) + 1 \cdot 1}{31}$

Step 4.3.1.2.5

Evaluate the exponent.

$\frac{16 \cdot 2 + 4 \sqrt{2} \cdot 1 + 1 (4 \sqrt{2}) + 1 \cdot 1}{31}$

Step 4.3.1.3

Multiply $16$ by $2$ .

$\frac{32 + 4 \sqrt{2} \cdot 1 + 1 (4 \sqrt{2}) + 1 \cdot 1}{31}$

Step 4.3.1.4

Multiply $4$ by $1$ .

$\frac{32 + 4 \sqrt{2} + 1 (4 \sqrt{2}) + 1 \cdot 1}{31}$

Step 4.3.1.5

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

$\frac{32 + 4 \sqrt{2} + 4 \sqrt{2} + 1 \cdot 1}{31}$

Step 4.3.1.6

Multiply $1$ by $1$ .

$\frac{32 + 4 \sqrt{2} + 4 \sqrt{2} + 1}{31}$

Step 4.3.2

Add $32$ and $1$ .

$\frac{33 + 4 \sqrt{2} + 4 \sqrt{2}}{31}$

Step 4.3.3

Add $4 \sqrt{2}$ and $4 \sqrt{2}$ .

$\frac{33 + 8 \sqrt{2}}{31}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{33 + 8 \sqrt{2}}{31}$

Decimal Form:

$1.42947446 \dots$