Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $2$ by $2$ .

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

Step 1.3.1.2

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

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

Step 1.3.1.3

Combine using the product rule for radicals.

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

Step 1.3.1.4

Multiply $3$ by $3$ .

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

Step 1.3.1.5

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

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

Step 1.3.1.6

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

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

Step 1.3.2

Add $4$ and $3$ .

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

Step 1.3.3

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

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

Step 1.4

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

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

Step 1.5

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

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Step 1.5.1

Apply the distributive property.

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

Step 1.5.2

Apply the distributive property.

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

Step 1.5.3

Apply the distributive property.

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

Step 1.6

Simplify and combine like terms.

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Step 1.6.1

Simplify each term.

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Step 1.6.1.1

Multiply $2$ by $2$ .

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

Step 1.6.1.2

Multiply $- 1$ by $2$ .

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

Step 1.6.1.3

Multiply $2$ by $- 1$ .

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

Step 1.6.1.4

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

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Step 1.6.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 1.6.1.4.2

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

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

Step 1.6.1.4.3

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

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

Step 1.6.1.4.4

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

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

Step 1.6.1.4.5

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

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

Step 1.6.1.4.6

Add $1$ and $1$ .

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

Step 1.6.1.5

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

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Step 1.6.1.5.1

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

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

Step 1.6.1.5.2

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

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

Step 1.6.1.5.3

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

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

Step 1.6.1.5.4

Cancel the common factor of $2$ .

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Step 1.6.1.5.4.1

Cancel the common factor.

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

Step 1.6.1.5.4.2

Rewrite the expression.

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

Step 1.6.1.5.5

Evaluate the exponent.

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

Step 1.6.2

Add $4$ and $3$ .

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

Step 1.6.3

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

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

Step 1.7

Apply the distributive property.

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

Step 1.8

Multiply $- 1$ by $7$ .

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

Step 1.9

Multiply $- 4$ by $- 1$ .

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

Step 2

Simplify by adding terms.

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Step 2.1

Subtract $7$ from $7$ .

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

Step 2.2

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

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

Step 2.3

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:

$13.85640646 \dots$