Basic Math Examples

$(3 \sqrt{2 - 2 \sqrt{3}}) (3 \sqrt{2 + 2 \sqrt{3}})$

Step 1

Multiply $(3 \sqrt{2 - 2 \sqrt{3}}) (3 \sqrt{2 + 2 \sqrt{3}})$ .

Tap for more steps...

Step 1.1

Multiply $3$ by $3$ .

$9 \sqrt{2 - 2 \sqrt{3}} \sqrt{2 + 2 \sqrt{3}}$

Step 1.2

Combine using the product rule for radicals.

$9 \sqrt{(2 + 2 \sqrt{3}) (2 - 2 \sqrt{3})}$

Step 1.3

Expand $(2 + 2 \sqrt{3}) (2 - 2 \sqrt{3})$ using the FOIL Method.

Tap for more steps...

Step 1.3.1

Apply the distributive property.

$9 \sqrt{2 (2 - 2 \sqrt{3}) + 2 \sqrt{3} (2 - 2 \sqrt{3})}$

Step 1.3.2

Apply the distributive property.

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

Step 1.3.3

Apply the distributive property.

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

Step 1.4

Simplify and combine like terms.

Tap for more steps...

Step 1.4.1

Simplify each term.

Tap for more steps...

Step 1.4.1.1

Multiply $2$ by $2$ .

$9 \sqrt{4 + 2 (- 2 \sqrt{3}) + 2 \sqrt{3} \cdot 2 + 2 \sqrt{3} (- 2 \sqrt{3})}$

Step 1.4.1.2

Multiply $- 2$ by $2$ .

$9 \sqrt{4 - 4 \sqrt{3} + 2 \sqrt{3} \cdot 2 + 2 \sqrt{3} (- 2 \sqrt{3})}$

Step 1.4.1.3

Multiply $2$ by $2$ .

$9 \sqrt{4 - 4 \sqrt{3} + 4 \sqrt{3} + 2 \sqrt{3} (- 2 \sqrt{3})}$

Step 1.4.1.4

Multiply $2 \sqrt{3} (- 2 \sqrt{3})$ .

Tap for more steps...

Step 1.4.1.4.1

Multiply $- 2$ by $2$ .

$9 \sqrt{4 - 4 \sqrt{3} + 4 \sqrt{3} - 4 \sqrt{3} \sqrt{3}}$

Step 1.4.1.4.2

Raise $\sqrt{3}$ to the power of $1$ .

$9 \sqrt{4 - 4 \sqrt{3} + 4 \sqrt{3} - 4 ({\sqrt{3}}^{1} \sqrt{3})}$

Step 1.4.1.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$9 \sqrt{4 - 4 \sqrt{3} + 4 \sqrt{3} - 4 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}$

Step 1.4.1.4.4

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

$9 \sqrt{4 - 4 \sqrt{3} + 4 \sqrt{3} - 4 {\sqrt{3}}^{1 + 1}}$

Step 1.4.1.4.5

Add $1$ and $1$ .

$9 \sqrt{4 - 4 \sqrt{3} + 4 \sqrt{3} - 4 {\sqrt{3}}^{2}}$

Step 1.4.1.5

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Tap for more steps...

Step 1.4.1.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$9 \sqrt{4 - 4 \sqrt{3} + 4 \sqrt{3} - 4 {(3^{\frac{1}{2}})}^{2}}$

Step 1.4.1.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$9 \sqrt{4 - 4 \sqrt{3} + 4 \sqrt{3} - 4 \cdot 3^{\frac{1}{2} \cdot 2}}$

Step 1.4.1.5.3

Combine $\frac{1}{2}$ and $2$ .

$9 \sqrt{4 - 4 \sqrt{3} + 4 \sqrt{3} - 4 \cdot 3^{\frac{2}{2}}}$

Step 1.4.1.5.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.4.1.5.4.1

Cancel the common factor.

$9 \sqrt{4 - 4 \sqrt{3} + 4 \sqrt{3} - 4 \cdot 3^{\frac{2}{2}}}$

Step 1.4.1.5.4.2

Rewrite the expression.

$9 \sqrt{4 - 4 \sqrt{3} + 4 \sqrt{3} - 4 \cdot 3^{1}}$

Step 1.4.1.5.5

Evaluate the exponent.

$9 \sqrt{4 - 4 \sqrt{3} + 4 \sqrt{3} - 4 \cdot 3}$

Step 1.4.1.6

Multiply $- 4$ by $3$ .

$9 \sqrt{4 - 4 \sqrt{3} + 4 \sqrt{3} - 12}$

Step 1.4.2

Subtract $12$ from $4$ .

$9 \sqrt{- 8 - 4 \sqrt{3} + 4 \sqrt{3}}$

Step 1.4.3

Add $- 4 \sqrt{3}$ and $4 \sqrt{3}$ .

$9 \sqrt{- 8 + 0}$

Step 1.4.4

Add $- 8$ and $0$ .

$9 \sqrt{- 8}$

Step 2

Rewrite $- 8$ as $- 1 (8)$ .

$9 \sqrt{- 1 (8)}$

Step 3

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$9 (\sqrt{- 1} \cdot \sqrt{8})$

Step 4

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

$9 (i \cdot \sqrt{8})$

Step 5

Rewrite $8$ as $2^{2} \cdot 2$ .

Tap for more steps...

Step 5.1

Factor $4$ out of $8$ .

$9 (i \cdot \sqrt{4 (2)})$

Step 5.2

Rewrite $4$ as $2^{2}$ .

$9 (i \cdot \sqrt{2^{2} \cdot 2})$

Step 6

Pull terms out from under the radical.

$9 (i \cdot (2 \sqrt{2}))$

Step 7

Simplify the expression.

Tap for more steps...

Step 7.1

Move $2$ to the left of $i$ .

$9 (2 \cdot i \sqrt{2})$

Step 7.2

Multiply $2$ by $9$ .

$18 i \sqrt{2}$