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

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

Step 1

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 2

Simplify and combine like terms.

Simplify each term.

Combine using the product rule for radicals.

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

Multiply $7$ by $7$ .

$\sqrt{49} + \sqrt{7} (- i) + i \sqrt{7} + i (- i)$

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

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

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

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

Multiply $i (- i)$ .

Raise $i$ to the power of $1$ .

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

Raise $i$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

Rewrite $i^{2}$ as $- 1$ .

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

Multiply $- 1$ by $- 1$ .

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

Add $7$ and $1$ .

$\sqrt{7} (- i) + i \sqrt{7} + 8$

Reorder the factors of $\sqrt{7} (- i)$ .

$- i \sqrt{7} + i \sqrt{7} + 8$

Add $- i \sqrt{7}$ and $i \sqrt{7}$ .

$0 + 8$

Add $0$ and $8$ .

$8$