Simplify (2+ square root of 3)(2- square root of 3)

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

Step 1

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 2

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $2$ .

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

Multiply $- 1$ by $2$ .

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

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

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Multiply $- 1$ by $3$ .

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

Subtract $3$ from $4$ .

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

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

$1 + 0$

Add $1$ and $0$ .

$1$

Simplify (2+ square root of 3)(2- square root of 3)

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