Basic Math Examples

$(\sqrt{6} - \sqrt{- 9}) (\sqrt{6} + \sqrt{- 9})$

Step 1

Rewrite $- 9$ as $- 1 (9)$ .

$(\sqrt{6} - \sqrt{- 1 (9)}) (\sqrt{6} + \sqrt{- 9})$

Step 2

Rewrite $\sqrt{- 1 (9)}$ as $\sqrt{- 1} \cdot \sqrt{9}$ .

$(\sqrt{6} - (\sqrt{- 1} \cdot \sqrt{9})) (\sqrt{6} + \sqrt{- 9})$

Step 3

Rewrite $\sqrt{- 1}$ as $i$ .

$(\sqrt{6} - (i \cdot \sqrt{9})) (\sqrt{6} + \sqrt{- 9})$

Step 4

Rewrite $- 9$ as $- 1 (9)$ .

$(\sqrt{6} - (i \cdot \sqrt{9})) (\sqrt{6} + \sqrt{- 1 (9)})$

Step 5

Rewrite $\sqrt{- 1 (9)}$ as $\sqrt{- 1} \cdot \sqrt{9}$ .

$(\sqrt{6} - (i \cdot \sqrt{9})) (\sqrt{6} + \sqrt{- 1} \cdot \sqrt{9})$

Step 6

Rewrite $\sqrt{- 1}$ as $i$ .

$(\sqrt{6} - (i \cdot \sqrt{9})) (\sqrt{6} + i \cdot \sqrt{9})$

Step 7

Simplify terms.

Tap for more steps...

Step 7.1

Simplify each term.

Tap for more steps...

Step 7.1.1

Rewrite $9$ as $3^{2}$ .

$(\sqrt{6} - (i \cdot \sqrt{3^{2}})) (\sqrt{6} + i \cdot \sqrt{9})$

Step 7.1.2

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

$(\sqrt{6} - (i \cdot 3)) (\sqrt{6} + i \cdot \sqrt{9})$

Step 7.1.3

Move $3$ to the left of $i$ .

$(\sqrt{6} - (3 \cdot i)) (\sqrt{6} + i \cdot \sqrt{9})$

Step 7.1.4

Multiply $3$ by $- 1$ .

$(\sqrt{6} - 3 i) (\sqrt{6} + i \cdot \sqrt{9})$

Step 7.2

Simplify each term.

Tap for more steps...

Step 7.2.1

Rewrite $9$ as $3^{2}$ .

$(\sqrt{6} - 3 i) (\sqrt{6} + i \cdot \sqrt{3^{2}})$

Step 7.2.2

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

$(\sqrt{6} - 3 i) (\sqrt{6} + i \cdot 3)$

Step 7.2.3

Move $3$ to the left of $i$ .

$(\sqrt{6} - 3 i) (\sqrt{6} + 3 i)$

Step 8

Expand $(\sqrt{6} - 3 i) (\sqrt{6} + 3 i)$ using the FOIL Method.

Tap for more steps...

Step 8.1

Apply the distributive property.

$\sqrt{6} (\sqrt{6} + 3 i) - 3 i (\sqrt{6} + 3 i)$

Step 8.2

Apply the distributive property.

$\sqrt{6} \sqrt{6} + \sqrt{6} (3 i) - 3 i (\sqrt{6} + 3 i)$

Step 8.3

Apply the distributive property.

$\sqrt{6} \sqrt{6} + \sqrt{6} (3 i) - 3 i \sqrt{6} - 3 i (3 i)$

Step 9

Simplify and combine like terms.

Tap for more steps...

Step 9.1

Simplify each term.

Tap for more steps...

Step 9.1.1

Combine using the product rule for radicals.

$\sqrt{6 \cdot 6} + \sqrt{6} (3 i) - 3 i \sqrt{6} - 3 i (3 i)$

Step 9.1.2

Multiply $6$ by $6$ .

$\sqrt{36} + \sqrt{6} (3 i) - 3 i \sqrt{6} - 3 i (3 i)$

Step 9.1.3

Rewrite $36$ as $6^{2}$ .

$\sqrt{6^{2}} + \sqrt{6} (3 i) - 3 i \sqrt{6} - 3 i (3 i)$

Step 9.1.4

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

$6 + \sqrt{6} (3 i) - 3 i \sqrt{6} - 3 i (3 i)$

Step 9.1.5

Move $3$ to the left of $\sqrt{6}$ .

$6 + 3 \cdot \sqrt{6} i - 3 i \sqrt{6} - 3 i (3 i)$

Step 9.1.6

Multiply $- 3 i (3 i)$ .

Tap for more steps...

Step 9.1.6.1

Multiply $3$ by $- 3$ .

$6 + 3 \sqrt{6} i - 3 i \sqrt{6} - 9 i i$

Step 9.1.6.2

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

$6 + 3 \sqrt{6} i - 3 i \sqrt{6} - 9 (i^{1} i)$

Step 9.1.6.3

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

$6 + 3 \sqrt{6} i - 3 i \sqrt{6} - 9 (i^{1} i^{1})$

Step 9.1.6.4

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

$6 + 3 \sqrt{6} i - 3 i \sqrt{6} - 9 i^{1 + 1}$

Step 9.1.6.5

Add $1$ and $1$ .

$6 + 3 \sqrt{6} i - 3 i \sqrt{6} - 9 i^{2}$

Step 9.1.7

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

$6 + 3 \sqrt{6} i - 3 i \sqrt{6} - 9 \cdot - 1$

Step 9.1.8

Multiply $- 9$ by $- 1$ .

$6 + 3 \sqrt{6} i - 3 i \sqrt{6} + 9$

Step 9.2

Add $6$ and $9$ .

$3 \sqrt{6} i - 3 i \sqrt{6} + 15$

Step 9.3

Reorder the factors of $3 \sqrt{6} i$ .

$3 i \sqrt{6} - 3 i \sqrt{6} + 15$

Step 9.4

Subtract $3 i \sqrt{6}$ from $3 i \sqrt{6}$ .

$0 + 15$

Step 9.5

Add $0$ and $15$ .

$15$