Simplify (2- square root of 7)(1+2 square root of 7)

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

Step 1

Expand $(2 - \sqrt{7}) (1 + 2 \sqrt{7})$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 2

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $1$ .

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

Multiply $2$ by $2$ .

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

Multiply $- 1$ by $1$ .

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

Multiply $- \sqrt{7} (2 \sqrt{7})$ .

Multiply $2$ by $- 1$ .

$2 + 4 \sqrt{7} - \sqrt{7} - 2 \sqrt{7} \sqrt{7}$

Raise $\sqrt{7}$ to the power of $1$ .

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

Raise $\sqrt{7}$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

$2 + 4 \sqrt{7} - \sqrt{7} - 2 {\sqrt{7}}^{2}$

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

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

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

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

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

$2 + 4 \sqrt{7} - \sqrt{7} - 2 \cdot 7^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$2 + 4 \sqrt{7} - \sqrt{7} - 2 \cdot 7^{\frac{2}{2}}$

Rewrite the expression.

$2 + 4 \sqrt{7} - \sqrt{7} - 2 \cdot 7^{1}$

Evaluate the exponent.

$2 + 4 \sqrt{7} - \sqrt{7} - 2 \cdot 7$

Multiply $- 2$ by $7$ .

$2 + 4 \sqrt{7} - \sqrt{7} - 14$

Subtract $14$ from $2$ .

$- 12 + 4 \sqrt{7} - \sqrt{7}$

Subtract $\sqrt{7}$ from $4 \sqrt{7}$ .

$- 12 + 3 \sqrt{7}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$- 12 + 3 \sqrt{7}$

Decimal Form:

$- 4.06274606 \dots$

Simplify (2- square root of 7)(1+2 square root of 7)

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