Simplify square root of 4+ square root of 7* square root of 4- square root of 7

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

Combine using the product rule for radicals.

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

Expand $(4 + \sqrt{7}) (4 - \sqrt{7})$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $4$ by $4$ .

$\sqrt{16 + 4 (- \sqrt{7}) + \sqrt{7} \cdot 4 + \sqrt{7} (- \sqrt{7})}$

Multiply $- 1$ by $4$ .

$\sqrt{16 - 4 \sqrt{7} + \sqrt{7} \cdot 4 + \sqrt{7} (- \sqrt{7})}$

Move $4$ to the left of $\sqrt{7}$ .

$\sqrt{16 - 4 \sqrt{7} + 4 \cdot \sqrt{7} + \sqrt{7} (- \sqrt{7})}$

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

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

$\sqrt{16 - 4 \sqrt{7} + 4 \sqrt{7} - ({\sqrt{7}}^{1} \sqrt{7})}$

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

$\sqrt{16 - 4 \sqrt{7} + 4 \sqrt{7} - ({\sqrt{7}}^{1} {\sqrt{7}}^{1})}$

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

$\sqrt{16 - 4 \sqrt{7} + 4 \sqrt{7} - {\sqrt{7}}^{1 + 1}}$

Add $1$ and $1$ .

$\sqrt{16 - 4 \sqrt{7} + 4 \sqrt{7} - {\sqrt{7}}^{2}}$

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

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

$\sqrt{16 - 4 \sqrt{7} + 4 \sqrt{7} - {(7^{\frac{1}{2}})}^{2}}$

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

$\sqrt{16 - 4 \sqrt{7} + 4 \sqrt{7} - 7^{\frac{1}{2} \cdot 2}}$

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

$\sqrt{16 - 4 \sqrt{7} + 4 \sqrt{7} - 7^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sqrt{16 - 4 \sqrt{7} + 4 \sqrt{7} - 7^{\frac{2}{2}}}$

Rewrite the expression.

$\sqrt{16 - 4 \sqrt{7} + 4 \sqrt{7} - 7^{1}}$

Evaluate the exponent.

$\sqrt{16 - 4 \sqrt{7} + 4 \sqrt{7} - 1 \cdot 7}$

Multiply $- 1$ by $7$ .

$\sqrt{16 - 4 \sqrt{7} + 4 \sqrt{7} - 7}$

Subtract $7$ from $16$ .

$\sqrt{9 - 4 \sqrt{7} + 4 \sqrt{7}}$

Add $- 4 \sqrt{7}$ and $4 \sqrt{7}$ .

$\sqrt{9 + 0}$

Add $9$ and $0$ .

$\sqrt{9}$

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

$\sqrt{3^{2}}$

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

$3$

Simplify square root of 4+ square root of 7* square root of 4- square root of 7

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