Trigonometry Examples

$(4 - \sqrt{13}) (4 + \sqrt{13})$

Step 1

Expand $(4 - \sqrt{13}) (4 + \sqrt{13})$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$4 (4 + \sqrt{13}) - \sqrt{13} (4 + \sqrt{13})$

Step 1.2

Apply the distributive property.

$4 \cdot 4 + 4 \sqrt{13} - \sqrt{13} (4 + \sqrt{13})$

Step 1.3

Apply the distributive property.

$4 \cdot 4 + 4 \sqrt{13} - \sqrt{13} \cdot 4 - \sqrt{13} \sqrt{13}$

Step 2

Simplify and combine like terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply $4$ by $4$ .

$16 + 4 \sqrt{13} - \sqrt{13} \cdot 4 - \sqrt{13} \sqrt{13}$

Step 2.1.2

Multiply $4$ by $- 1$ .

$16 + 4 \sqrt{13} - 4 \sqrt{13} - \sqrt{13} \sqrt{13}$

Step 2.1.3

Multiply $- \sqrt{13} \sqrt{13}$ .

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Step 2.1.3.1

Raise $\sqrt{13}$ to the power of $1$ .

$16 + 4 \sqrt{13} - 4 \sqrt{13} - ({\sqrt{13}}^{1} \sqrt{13})$

Step 2.1.3.2

Raise $\sqrt{13}$ to the power of $1$ .

$16 + 4 \sqrt{13} - 4 \sqrt{13} - ({\sqrt{13}}^{1} {\sqrt{13}}^{1})$

Step 2.1.3.3

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

$16 + 4 \sqrt{13} - 4 \sqrt{13} - {\sqrt{13}}^{1 + 1}$

Step 2.1.3.4

Add $1$ and $1$ .

$16 + 4 \sqrt{13} - 4 \sqrt{13} - {\sqrt{13}}^{2}$

Step 2.1.4

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

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Step 2.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$16 + 4 \sqrt{13} - 4 \sqrt{13} - {(13^{\frac{1}{2}})}^{2}$

Step 2.1.4.2

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

$16 + 4 \sqrt{13} - 4 \sqrt{13} - 13^{\frac{1}{2} \cdot 2}$

Step 2.1.4.3

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

$16 + 4 \sqrt{13} - 4 \sqrt{13} - 13^{\frac{2}{2}}$

Step 2.1.4.4

Cancel the common factor of $2$ .

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Step 2.1.4.4.1

Cancel the common factor.

$16 + 4 \sqrt{13} - 4 \sqrt{13} - 13^{\frac{2}{2}}$

Step 2.1.4.4.2

Rewrite the expression.

$16 + 4 \sqrt{13} - 4 \sqrt{13} - 13^{1}$

Step 2.1.4.5

Evaluate the exponent.

$16 + 4 \sqrt{13} - 4 \sqrt{13} - 1 \cdot 13$

Step 2.1.5

Multiply $- 1$ by $13$ .

$16 + 4 \sqrt{13} - 4 \sqrt{13} - 13$

Step 2.2

Subtract $13$ from $16$ .

$3 + 4 \sqrt{13} - 4 \sqrt{13}$

Step 2.3

Subtract $4 \sqrt{13}$ from $4 \sqrt{13}$ .

$3 + 0$

Step 2.4

Add $3$ and $0$ .

$3$