Finite Math Examples

$(\sqrt{5} + 1) (3 \sqrt{5} + 1)$

Step 1

Expand $(\sqrt{5} + 1) (3 \sqrt{5} + 1)$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$\sqrt{5} (3 \sqrt{5} + 1) + 1 (3 \sqrt{5} + 1)$

Step 1.2

Apply the distributive property.

$\sqrt{5} (3 \sqrt{5}) + \sqrt{5} \cdot 1 + 1 (3 \sqrt{5} + 1)$

Step 1.3

Apply the distributive property.

$\sqrt{5} (3 \sqrt{5}) + \sqrt{5} \cdot 1 + 1 (3 \sqrt{5}) + 1 \cdot 1$

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 $\sqrt{5} (3 \sqrt{5})$ .

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Step 2.1.1.1

Raise $\sqrt{5}$ to the power of $1$ .

$3 ({\sqrt{5}}^{1} \sqrt{5}) + \sqrt{5} \cdot 1 + 1 (3 \sqrt{5}) + 1 \cdot 1$

Step 2.1.1.2

Raise $\sqrt{5}$ to the power of $1$ .

$3 ({\sqrt{5}}^{1} {\sqrt{5}}^{1}) + \sqrt{5} \cdot 1 + 1 (3 \sqrt{5}) + 1 \cdot 1$

Step 2.1.1.3

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

$3 {\sqrt{5}}^{1 + 1} + \sqrt{5} \cdot 1 + 1 (3 \sqrt{5}) + 1 \cdot 1$

Step 2.1.1.4

Add $1$ and $1$ .

$3 {\sqrt{5}}^{2} + \sqrt{5} \cdot 1 + 1 (3 \sqrt{5}) + 1 \cdot 1$

Step 2.1.2

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 2.1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$3 {(5^{\frac{1}{2}})}^{2} + \sqrt{5} \cdot 1 + 1 (3 \sqrt{5}) + 1 \cdot 1$

Step 2.1.2.2

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

$3 \cdot 5^{\frac{1}{2} \cdot 2} + \sqrt{5} \cdot 1 + 1 (3 \sqrt{5}) + 1 \cdot 1$

Step 2.1.2.3

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

$3 \cdot 5^{\frac{2}{2}} + \sqrt{5} \cdot 1 + 1 (3 \sqrt{5}) + 1 \cdot 1$

Step 2.1.2.4

Cancel the common factor of $2$ .

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Step 2.1.2.4.1

Cancel the common factor.

$3 \cdot 5^{\frac{2}{2}} + \sqrt{5} \cdot 1 + 1 (3 \sqrt{5}) + 1 \cdot 1$

Step 2.1.2.4.2

Rewrite the expression.

$3 \cdot 5^{1} + \sqrt{5} \cdot 1 + 1 (3 \sqrt{5}) + 1 \cdot 1$

Step 2.1.2.5

Evaluate the exponent.

$3 \cdot 5 + \sqrt{5} \cdot 1 + 1 (3 \sqrt{5}) + 1 \cdot 1$

Step 2.1.3

Multiply $3$ by $5$ .

$15 + \sqrt{5} \cdot 1 + 1 (3 \sqrt{5}) + 1 \cdot 1$

Step 2.1.4

Multiply $\sqrt{5}$ by $1$ .

$15 + \sqrt{5} + 1 (3 \sqrt{5}) + 1 \cdot 1$

Step 2.1.5

Multiply $3 \sqrt{5}$ by $1$ .

$15 + \sqrt{5} + 3 \sqrt{5} + 1 \cdot 1$

Step 2.1.6

Multiply $1$ by $1$ .

$15 + \sqrt{5} + 3 \sqrt{5} + 1$

Step 2.2

Add $15$ and $1$ .

$16 + \sqrt{5} + 3 \sqrt{5}$

Step 2.3

Add $\sqrt{5}$ and $3 \sqrt{5}$ .

$16 + 4 \sqrt{5}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$16 + 4 \sqrt{5}$

Decimal Form:

$24.94427190 \dots$