Basic Math Examples

$(1 + \sqrt{2}) (3 - \sqrt{2})$

Step 1

Expand $(1 + \sqrt{2}) (3 - \sqrt{2})$ using the FOIL Method.

Tap for more steps...

Step 1.1

Apply the distributive property.

$1 (3 - \sqrt{2}) + \sqrt{2} (3 - \sqrt{2})$

Step 1.2

Apply the distributive property.

$1 \cdot 3 + 1 (- \sqrt{2}) + \sqrt{2} (3 - \sqrt{2})$

Step 1.3

Apply the distributive property.

$1 \cdot 3 + 1 (- \sqrt{2}) + \sqrt{2} \cdot 3 + \sqrt{2} (- \sqrt{2})$

Step 2

Simplify and combine like terms.

Tap for more steps...

Step 2.1

Simplify each term.

Tap for more steps...

Step 2.1.1

Multiply $3$ by $1$ .

$3 + 1 (- \sqrt{2}) + \sqrt{2} \cdot 3 + \sqrt{2} (- \sqrt{2})$

Step 2.1.2

Multiply $- \sqrt{2}$ by $1$ .

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

Step 2.1.3

Move $3$ to the left of $\sqrt{2}$ .

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

Step 2.1.4

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

Tap for more steps...

Step 2.1.4.1

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 2.1.4.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 2.1.4.3

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

$3 - \sqrt{2} + 3 \sqrt{2} - {\sqrt{2}}^{1 + 1}$

Step 2.1.4.4

Add $1$ and $1$ .

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

Step 2.1.5

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

Tap for more steps...

Step 2.1.5.1

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

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

Step 2.1.5.2

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

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

Step 2.1.5.3

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

$3 - \sqrt{2} + 3 \sqrt{2} - 2^{\frac{2}{2}}$

Step 2.1.5.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.1.5.4.1

Cancel the common factor.

$3 - \sqrt{2} + 3 \sqrt{2} - 2^{\frac{2}{2}}$

Step 2.1.5.4.2

Rewrite the expression.

$3 - \sqrt{2} + 3 \sqrt{2} - 2^{1}$

Step 2.1.5.5

Evaluate the exponent.

$3 - \sqrt{2} + 3 \sqrt{2} - 1 \cdot 2$

Step 2.1.6

Multiply $- 1$ by $2$ .

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

Step 2.2

Subtract $2$ from $3$ .

$1 - \sqrt{2} + 3 \sqrt{2}$

Step 2.3

Add $- \sqrt{2}$ and $3 \sqrt{2}$ .

$1 + 2 \sqrt{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$1 + 2 \sqrt{2}$

Decimal Form:

$3.82842712 \dots$