Basic Math Examples

$(7 + 4 \sqrt{5}) \cdot (7 + \sqrt{5})$

Step 1

Expand $(7 + 4 \sqrt{5}) (7 + \sqrt{5})$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$7 (7 + \sqrt{5}) + 4 \sqrt{5} (7 + \sqrt{5})$

Step 1.2

Apply the distributive property.

$7 \cdot 7 + 7 \sqrt{5} + 4 \sqrt{5} (7 + \sqrt{5})$

Step 1.3

Apply the distributive property.

$7 \cdot 7 + 7 \sqrt{5} + 4 \sqrt{5} \cdot 7 + 4 \sqrt{5} \sqrt{5}$

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 $7$ by $7$ .

$49 + 7 \sqrt{5} + 4 \sqrt{5} \cdot 7 + 4 \sqrt{5} \sqrt{5}$

Step 2.1.2

Multiply $7$ by $4$ .

$49 + 7 \sqrt{5} + 28 \sqrt{5} + 4 \sqrt{5} \sqrt{5}$

Step 2.1.3

Multiply $4 \sqrt{5} \sqrt{5}$ .

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Step 2.1.3.1

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

$49 + 7 \sqrt{5} + 28 \sqrt{5} + 4 ({\sqrt{5}}^{1} \sqrt{5})$

Step 2.1.3.2

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

$49 + 7 \sqrt{5} + 28 \sqrt{5} + 4 ({\sqrt{5}}^{1} {\sqrt{5}}^{1})$

Step 2.1.3.3

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

$49 + 7 \sqrt{5} + 28 \sqrt{5} + 4 {\sqrt{5}}^{1 + 1}$

Step 2.1.3.4

Add $1$ and $1$ .

$49 + 7 \sqrt{5} + 28 \sqrt{5} + 4 {\sqrt{5}}^{2}$

Step 2.1.4

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

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Step 2.1.4.1

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

$49 + 7 \sqrt{5} + 28 \sqrt{5} + 4 {(5^{\frac{1}{2}})}^{2}$

Step 2.1.4.2

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

$49 + 7 \sqrt{5} + 28 \sqrt{5} + 4 \cdot 5^{\frac{1}{2} \cdot 2}$

Step 2.1.4.3

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

$49 + 7 \sqrt{5} + 28 \sqrt{5} + 4 \cdot 5^{\frac{2}{2}}$

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.

$49 + 7 \sqrt{5} + 28 \sqrt{5} + 4 \cdot 5^{\frac{2}{2}}$

Step 2.1.4.4.2

Rewrite the expression.

$49 + 7 \sqrt{5} + 28 \sqrt{5} + 4 \cdot 5^{1}$

Step 2.1.4.5

Evaluate the exponent.

$49 + 7 \sqrt{5} + 28 \sqrt{5} + 4 \cdot 5$

Step 2.1.5

Multiply $4$ by $5$ .

$49 + 7 \sqrt{5} + 28 \sqrt{5} + 20$

Step 2.2

Add $49$ and $20$ .

$69 + 7 \sqrt{5} + 28 \sqrt{5}$

Step 2.3

Add $7 \sqrt{5}$ and $28 \sqrt{5}$ .

$69 + 35 \sqrt{5}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$69 + 35 \sqrt{5}$

Decimal Form:

$147.26237921 \dots$