Simplify the Radical Expression ( square root of 2+ square root of 3)( square root of 2- square root of 3)

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

Step 1

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 2

Simplify and combine like terms.

Simplify each term.

Combine using the product rule for radicals.

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

Multiply $2$ by $2$ .

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

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

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

Pull terms out from under the radical.

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

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

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

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

Combine using the product rule for radicals.

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

Multiply $2$ by $3$ .

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

Combine using the product rule for radicals.

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

Multiply $3$ by $2$ .

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

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

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

$2 - \sqrt{6} + \sqrt{6} - ({\sqrt{3}}^{1} \sqrt{3})$

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

$2 - \sqrt{6} + \sqrt{6} - ({\sqrt{3}}^{1} {\sqrt{3}}^{1})$

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

$2 - \sqrt{6} + \sqrt{6} - {\sqrt{3}}^{1 + 1}$

Add $1$ and $1$ .

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

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

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

$2 - \sqrt{6} + \sqrt{6} - {(3^{\frac{1}{2}})}^{2}$

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

$2 - \sqrt{6} + \sqrt{6} - 3^{\frac{1}{2} \cdot 2}$

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

$2 - \sqrt{6} + \sqrt{6} - 3^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$2 - \sqrt{6} + \sqrt{6} - 3^{\frac{2}{2}}$

Rewrite the expression.

$2 - \sqrt{6} + \sqrt{6} - 3^{1}$

Evaluate the exponent.

$2 - \sqrt{6} + \sqrt{6} - 1 \cdot 3$

Multiply $- 1$ by $3$ .

$2 - \sqrt{6} + \sqrt{6} - 3$

Subtract $3$ from $2$ .

$- 1 - \sqrt{6} + \sqrt{6}$

Add $- \sqrt{6}$ and $\sqrt{6}$ .

$- 1 + 0$

Add $- 1$ and $0$ .

$- 1$

Simplify the Radical Expression ( square root of 2+ square root of 3)( square root of 2- square root of 3)

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