Simplify ( square root of 7-7)(1+ square root of 7)

$(\sqrt{7} - 7) (1 + \sqrt{7})$

Step 1

Expand $(\sqrt{7} - 7) (1 + \sqrt{7})$ using the FOIL Method.

Apply the distributive property.

$\sqrt{7} (1 + \sqrt{7}) - 7 (1 + \sqrt{7})$

Apply the distributive property.

$\sqrt{7} \cdot 1 + \sqrt{7} \sqrt{7} - 7 (1 + \sqrt{7})$

Apply the distributive property.

$\sqrt{7} \cdot 1 + \sqrt{7} \sqrt{7} - 7 \cdot 1 - 7 \sqrt{7}$

Step 2

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt{7}$ by $1$ .

$\sqrt{7} + \sqrt{7} \sqrt{7} - 7 \cdot 1 - 7 \sqrt{7}$

Combine using the product rule for radicals.

$\sqrt{7} + \sqrt{7 \cdot 7} - 7 \cdot 1 - 7 \sqrt{7}$

Multiply $7$ by $7$ .

$\sqrt{7} + \sqrt{49} - 7 \cdot 1 - 7 \sqrt{7}$

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

$\sqrt{7} + \sqrt{7^{2}} - 7 \cdot 1 - 7 \sqrt{7}$

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

$\sqrt{7} + 7 - 7 \cdot 1 - 7 \sqrt{7}$

Multiply $- 7$ by $1$ .

$\sqrt{7} + 7 - 7 - 7 \sqrt{7}$

Subtract $7 \sqrt{7}$ from $\sqrt{7}$ .

$7 - 7 - 6 \sqrt{7}$

Subtract $7$ from $7$ .

$0 - 6 \sqrt{7}$

Subtract $6 \sqrt{7}$ from $0$ .

$- 6 \sqrt{7}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$- 6 \sqrt{7}$

Decimal Form:

$- 15.87450786 \dots$