Pre-Algebra Examples

$(\sqrt{7} + 4 i) (\sqrt{7} - 4 i)$

Step 1

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

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Step 1.1

Apply the distributive property.

$\sqrt{7} (\sqrt{7} - 4 i) + 4 i (\sqrt{7} - 4 i)$

Step 1.2

Apply the distributive property.

$\sqrt{7} \sqrt{7} + \sqrt{7} (- 4 i) + 4 i (\sqrt{7} - 4 i)$

Step 1.3

Apply the distributive property.

$\sqrt{7} \sqrt{7} + \sqrt{7} (- 4 i) + 4 i \sqrt{7} + 4 i (- 4 i)$

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

Combine using the product rule for radicals.

$\sqrt{7 \cdot 7} + \sqrt{7} (- 4 i) + 4 i \sqrt{7} + 4 i (- 4 i)$

Step 2.1.2

Multiply $7$ by $7$ .

$\sqrt{49} + \sqrt{7} (- 4 i) + 4 i \sqrt{7} + 4 i (- 4 i)$

Step 2.1.3

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

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

Step 2.1.4

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

$7 + \sqrt{7} (- 4 i) + 4 i \sqrt{7} + 4 i (- 4 i)$

Step 2.1.5

Multiply $4 i (- 4 i)$ .

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Step 2.1.5.1

Multiply $- 4$ by $4$ .

$7 + \sqrt{7} (- 4 i) + 4 i \sqrt{7} - 16 i i$

Step 2.1.5.2

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

$7 + \sqrt{7} (- 4 i) + 4 i \sqrt{7} - 16 (i^{1} i)$

Step 2.1.5.3

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

$7 + \sqrt{7} (- 4 i) + 4 i \sqrt{7} - 16 (i^{1} i^{1})$

Step 2.1.5.4

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

$7 + \sqrt{7} (- 4 i) + 4 i \sqrt{7} - 16 i^{1 + 1}$

Step 2.1.5.5

Add $1$ and $1$ .

$7 + \sqrt{7} (- 4 i) + 4 i \sqrt{7} - 16 i^{2}$

Step 2.1.6

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

$7 + \sqrt{7} (- 4 i) + 4 i \sqrt{7} - 16 \cdot - 1$

Step 2.1.7

Multiply $- 16$ by $- 1$ .

$7 + \sqrt{7} (- 4 i) + 4 i \sqrt{7} + 16$

Step 2.2

Add $7$ and $16$ .

$\sqrt{7} (- 4 i) + 4 i \sqrt{7} + 23$

Step 2.3

Reorder the factors of $\sqrt{7} (- 4 i)$ .

$- 4 i \sqrt{7} + 4 i \sqrt{7} + 23$

Step 2.4

Add $- 4 i \sqrt{7}$ and $4 i \sqrt{7}$ .

$0 + 23$

Step 2.5

Add $0$ and $23$ .

$23$