Calculus Examples

$- 32 {(2 + \sqrt{7})}^{2} + 64$

Step 1

Simplify each term.

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Step 1.1

Rewrite ${(2 + \sqrt{7})}^{2}$ as $(2 + \sqrt{7}) (2 + \sqrt{7})$ .

$- 32 ((2 + \sqrt{7}) (2 + \sqrt{7})) + 64$

Step 1.2

Expand $(2 + \sqrt{7}) (2 + \sqrt{7})$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$- 32 (2 (2 + \sqrt{7}) + \sqrt{7} (2 + \sqrt{7})) + 64$

Step 1.2.2

Apply the distributive property.

$- 32 (2 \cdot 2 + 2 \sqrt{7} + \sqrt{7} (2 + \sqrt{7})) + 64$

Step 1.2.3

Apply the distributive property.

$- 32 (2 \cdot 2 + 2 \sqrt{7} + \sqrt{7} \cdot 2 + \sqrt{7} \sqrt{7}) + 64$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $2$ by $2$ .

$- 32 (4 + 2 \sqrt{7} + \sqrt{7} \cdot 2 + \sqrt{7} \sqrt{7}) + 64$

Step 1.3.1.2

Move $2$ to the left of $\sqrt{7}$ .

$- 32 (4 + 2 \sqrt{7} + 2 \cdot \sqrt{7} + \sqrt{7} \sqrt{7}) + 64$

Step 1.3.1.3

Combine using the product rule for radicals.

$- 32 (4 + 2 \sqrt{7} + 2 \sqrt{7} + \sqrt{7 \cdot 7}) + 64$

Step 1.3.1.4

Multiply $7$ by $7$ .

$- 32 (4 + 2 \sqrt{7} + 2 \sqrt{7} + \sqrt{49}) + 64$

Step 1.3.1.5

Rewrite $49$ as $7^{2}$ .

$- 32 (4 + 2 \sqrt{7} + 2 \sqrt{7} + \sqrt{7^{2}}) + 64$

Step 1.3.1.6

Pull terms out from under the radical, assuming positive real numbers.

$- 32 (4 + 2 \sqrt{7} + 2 \sqrt{7} + 7) + 64$

Step 1.3.2

Add $4$ and $7$ .

$- 32 (11 + 2 \sqrt{7} + 2 \sqrt{7}) + 64$

Step 1.3.3

Add $2 \sqrt{7}$ and $2 \sqrt{7}$ .

$- 32 (11 + 4 \sqrt{7}) + 64$

Step 1.4

Apply the distributive property.

$- 32 \cdot 11 - 32 (4 \sqrt{7}) + 64$

Step 1.5

Multiply $- 32$ by $11$ .

$- 352 - 32 (4 \sqrt{7}) + 64$

Step 1.6

Multiply $4$ by $- 32$ .

$- 352 - 128 \sqrt{7} + 64$

Step 2

Add $- 352$ and $64$ .

$- 288 - 128 \sqrt{7}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$- 288 - 128 \sqrt{7}$

Decimal Form:

$- 626.65616781 \dots$