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

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

Step 1

Simplify each term.

Rewrite ${(2 + \sqrt{3})}^{2}$ as $(2 + \sqrt{3}) (2 + \sqrt{3})$ .

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $2$ .

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

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

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

Combine using the product rule for radicals.

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

Multiply $3$ by $3$ .

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

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

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

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

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

Add $4$ and $3$ .

$7 + 2 \sqrt{3} + 2 \sqrt{3} - {(2 - \sqrt{3})}^{2}$

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

$7 + 4 \sqrt{3} - {(2 - \sqrt{3})}^{2}$

Rewrite ${(2 - \sqrt{3})}^{2}$ as $(2 - \sqrt{3}) (2 - \sqrt{3})$ .

$7 + 4 \sqrt{3} - ((2 - \sqrt{3}) (2 - \sqrt{3}))$

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

Apply the distributive property.

$7 + 4 \sqrt{3} - (2 (2 - \sqrt{3}) - \sqrt{3} (2 - \sqrt{3}))$

Apply the distributive property.

$7 + 4 \sqrt{3} - (2 \cdot 2 + 2 (- \sqrt{3}) - \sqrt{3} (2 - \sqrt{3}))$

Apply the distributive property.

$7 + 4 \sqrt{3} - (2 \cdot 2 + 2 (- \sqrt{3}) - \sqrt{3} \cdot 2 - \sqrt{3} (- \sqrt{3}))$

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $2$ .

$7 + 4 \sqrt{3} - (4 + 2 (- \sqrt{3}) - \sqrt{3} \cdot 2 - \sqrt{3} (- \sqrt{3}))$

Multiply $- 1$ by $2$ .

$7 + 4 \sqrt{3} - (4 - 2 \sqrt{3} - \sqrt{3} \cdot 2 - \sqrt{3} (- \sqrt{3}))$

Multiply $2$ by $- 1$ .

$7 + 4 \sqrt{3} - (4 - 2 \sqrt{3} - 2 \sqrt{3} - \sqrt{3} (- \sqrt{3}))$

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

Multiply $- 1$ by $- 1$ .

$7 + 4 \sqrt{3} - (4 - 2 \sqrt{3} - 2 \sqrt{3} + 1 \sqrt{3} \sqrt{3})$

Multiply $\sqrt{3}$ by $1$ .

$7 + 4 \sqrt{3} - (4 - 2 \sqrt{3} - 2 \sqrt{3} + \sqrt{3} \sqrt{3})$

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

$7 + 4 \sqrt{3} - (4 - 2 \sqrt{3} - 2 \sqrt{3} + {\sqrt{3}}^{1} \sqrt{3})$

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

$7 + 4 \sqrt{3} - (4 - 2 \sqrt{3} - 2 \sqrt{3} + {\sqrt{3}}^{1} {\sqrt{3}}^{1})$

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

$7 + 4 \sqrt{3} - (4 - 2 \sqrt{3} - 2 \sqrt{3} + {\sqrt{3}}^{1 + 1})$

Add $1$ and $1$ .

$7 + 4 \sqrt{3} - (4 - 2 \sqrt{3} - 2 \sqrt{3} + {\sqrt{3}}^{2})$

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

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

$7 + 4 \sqrt{3} - (4 - 2 \sqrt{3} - 2 \sqrt{3} + {(3^{\frac{1}{2}})}^{2})$

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

$7 + 4 \sqrt{3} - (4 - 2 \sqrt{3} - 2 \sqrt{3} + 3^{\frac{1}{2} \cdot 2})$

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

$7 + 4 \sqrt{3} - (4 - 2 \sqrt{3} - 2 \sqrt{3} + 3^{\frac{2}{2}})$

Cancel the common factor of $2$ .

Cancel the common factor.

$7 + 4 \sqrt{3} - (4 - 2 \sqrt{3} - 2 \sqrt{3} + 3^{\frac{2}{2}})$

Rewrite the expression.

$7 + 4 \sqrt{3} - (4 - 2 \sqrt{3} - 2 \sqrt{3} + 3^{1})$

Evaluate the exponent.

$7 + 4 \sqrt{3} - (4 - 2 \sqrt{3} - 2 \sqrt{3} + 3)$

Add $4$ and $3$ .

$7 + 4 \sqrt{3} - (7 - 2 \sqrt{3} - 2 \sqrt{3})$

Subtract $2 \sqrt{3}$ from $- 2 \sqrt{3}$ .

$7 + 4 \sqrt{3} - (7 - 4 \sqrt{3})$

Apply the distributive property.

$7 + 4 \sqrt{3} - 1 \cdot 7 - (- 4 \sqrt{3})$

Multiply $- 1$ by $7$ .

$7 + 4 \sqrt{3} - 7 - (- 4 \sqrt{3})$

Multiply $- 4$ by $- 1$ .

$7 + 4 \sqrt{3} - 7 + 4 \sqrt{3}$

Step 2

Simplify by adding terms.

Subtract $7$ from $7$ .

$0 + 4 \sqrt{3} + 4 \sqrt{3}$

Add $0$ and $4 \sqrt{3}$ .

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

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

$8 \sqrt{3}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$8 \sqrt{3}$

Decimal Form:

$13.85640646 \dots$

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

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