Simplify ( square root of 10+2 square root of 8)( square root of 10-2 square root of 8)

$(\sqrt{10} + 2 \sqrt{8}) (\sqrt{10} - 2 \sqrt{8})$

Step 1

Simplify terms.

Simplify each term.

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

$(\sqrt{10} + 2 \sqrt{4 (2)}) (\sqrt{10} - 2 \sqrt{8})$

Rewrite $4$ as $2^{2}$ .

$(\sqrt{10} + 2 \sqrt{2^{2} \cdot 2}) (\sqrt{10} - 2 \sqrt{8})$

Pull terms out from under the radical.

$(\sqrt{10} + 2 (2 \sqrt{2})) (\sqrt{10} - 2 \sqrt{8})$

Multiply $2$ by $2$ .

$(\sqrt{10} + 4 \sqrt{2}) (\sqrt{10} - 2 \sqrt{8})$

Simplify each term.

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

$(\sqrt{10} + 4 \sqrt{2}) (\sqrt{10} - 2 \sqrt{4 (2)})$

Rewrite $4$ as $2^{2}$ .

$(\sqrt{10} + 4 \sqrt{2}) (\sqrt{10} - 2 \sqrt{2^{2} \cdot 2})$

Pull terms out from under the radical.

$(\sqrt{10} + 4 \sqrt{2}) (\sqrt{10} - 2 (2 \sqrt{2}))$

Multiply $2$ by $- 2$ .

$(\sqrt{10} + 4 \sqrt{2}) (\sqrt{10} - 4 \sqrt{2})$

Step 2

Expand $(\sqrt{10} + 4 \sqrt{2}) (\sqrt{10} - 4 \sqrt{2})$ using the FOIL Method.

Apply the distributive property.

$\sqrt{10} (\sqrt{10} - 4 \sqrt{2}) + 4 \sqrt{2} (\sqrt{10} - 4 \sqrt{2})$

Apply the distributive property.

$\sqrt{10} \sqrt{10} + \sqrt{10} (- 4 \sqrt{2}) + 4 \sqrt{2} (\sqrt{10} - 4 \sqrt{2})$

Apply the distributive property.

$\sqrt{10} \sqrt{10} + \sqrt{10} (- 4 \sqrt{2}) + 4 \sqrt{2} \sqrt{10} + 4 \sqrt{2} (- 4 \sqrt{2})$

Step 3

Simplify and combine like terms.

Simplify each term.

Combine using the product rule for radicals.

$\sqrt{10 \cdot 10} + \sqrt{10} (- 4 \sqrt{2}) + 4 \sqrt{2} \sqrt{10} + 4 \sqrt{2} (- 4 \sqrt{2})$

Multiply $10$ by $10$ .

$\sqrt{100} + \sqrt{10} (- 4 \sqrt{2}) + 4 \sqrt{2} \sqrt{10} + 4 \sqrt{2} (- 4 \sqrt{2})$

Rewrite $100$ as $10^{2}$ .

$\sqrt{10^{2}} + \sqrt{10} (- 4 \sqrt{2}) + 4 \sqrt{2} \sqrt{10} + 4 \sqrt{2} (- 4 \sqrt{2})$

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

$10 + \sqrt{10} (- 4 \sqrt{2}) + 4 \sqrt{2} \sqrt{10} + 4 \sqrt{2} (- 4 \sqrt{2})$

Multiply $\sqrt{10} (- 4 \sqrt{2})$ .

Combine using the product rule for radicals.

$10 - 4 \sqrt{10 \cdot 2} + 4 \sqrt{2} \sqrt{10} + 4 \sqrt{2} (- 4 \sqrt{2})$

Multiply $10$ by $2$ .

$10 - 4 \sqrt{20} + 4 \sqrt{2} \sqrt{10} + 4 \sqrt{2} (- 4 \sqrt{2})$

Rewrite $20$ as $2^{2} \cdot 5$ .

Factor $4$ out of $20$ .

$10 - 4 \sqrt{4 (5)} + 4 \sqrt{2} \sqrt{10} + 4 \sqrt{2} (- 4 \sqrt{2})$

Rewrite $4$ as $2^{2}$ .

$10 - 4 \sqrt{2^{2} \cdot 5} + 4 \sqrt{2} \sqrt{10} + 4 \sqrt{2} (- 4 \sqrt{2})$

Pull terms out from under the radical.

$10 - 4 (2 \sqrt{5}) + 4 \sqrt{2} \sqrt{10} + 4 \sqrt{2} (- 4 \sqrt{2})$

Multiply $2$ by $- 4$ .

$10 - 8 \sqrt{5} + 4 \sqrt{2} \sqrt{10} + 4 \sqrt{2} (- 4 \sqrt{2})$

Multiply $4 \sqrt{2} \sqrt{10}$ .

Combine using the product rule for radicals.

$10 - 8 \sqrt{5} + 4 \sqrt{10 \cdot 2} + 4 \sqrt{2} (- 4 \sqrt{2})$

Multiply $10$ by $2$ .

$10 - 8 \sqrt{5} + 4 \sqrt{20} + 4 \sqrt{2} (- 4 \sqrt{2})$

Rewrite $20$ as $2^{2} \cdot 5$ .

Factor $4$ out of $20$ .

$10 - 8 \sqrt{5} + 4 \sqrt{4 (5)} + 4 \sqrt{2} (- 4 \sqrt{2})$

Rewrite $4$ as $2^{2}$ .

$10 - 8 \sqrt{5} + 4 \sqrt{2^{2} \cdot 5} + 4 \sqrt{2} (- 4 \sqrt{2})$

Pull terms out from under the radical.

$10 - 8 \sqrt{5} + 4 (2 \sqrt{5}) + 4 \sqrt{2} (- 4 \sqrt{2})$

Multiply $2$ by $4$ .

$10 - 8 \sqrt{5} + 8 \sqrt{5} + 4 \sqrt{2} (- 4 \sqrt{2})$

Multiply $4 \sqrt{2} (- 4 \sqrt{2})$ .

Multiply $- 4$ by $4$ .

$10 - 8 \sqrt{5} + 8 \sqrt{5} - 16 \sqrt{2} \sqrt{2}$

Raise $\sqrt{2}$ to the power of $1$ .

$10 - 8 \sqrt{5} + 8 \sqrt{5} - 16 ({\sqrt{2}}^{1} \sqrt{2})$

Raise $\sqrt{2}$ to the power of $1$ .

$10 - 8 \sqrt{5} + 8 \sqrt{5} - 16 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})$

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

$10 - 8 \sqrt{5} + 8 \sqrt{5} - 16 {\sqrt{2}}^{1 + 1}$

Add $1$ and $1$ .

$10 - 8 \sqrt{5} + 8 \sqrt{5} - 16 {\sqrt{2}}^{2}$

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

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

$10 - 8 \sqrt{5} + 8 \sqrt{5} - 16 {(2^{\frac{1}{2}})}^{2}$

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

$10 - 8 \sqrt{5} + 8 \sqrt{5} - 16 \cdot 2^{\frac{1}{2} \cdot 2}$

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

$10 - 8 \sqrt{5} + 8 \sqrt{5} - 16 \cdot 2^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$10 - 8 \sqrt{5} + 8 \sqrt{5} - 16 \cdot 2^{\frac{2}{2}}$

Rewrite the expression.

$10 - 8 \sqrt{5} + 8 \sqrt{5} - 16 \cdot 2^{1}$

Evaluate the exponent.

$10 - 8 \sqrt{5} + 8 \sqrt{5} - 16 \cdot 2$

Multiply $- 16$ by $2$ .

$10 - 8 \sqrt{5} + 8 \sqrt{5} - 32$

Subtract $32$ from $10$ .

$- 22 - 8 \sqrt{5} + 8 \sqrt{5}$

Add $- 8 \sqrt{5}$ and $8 \sqrt{5}$ .

$- 22 + 0$

Add $- 22$ and $0$ .

$- 22$

Simplify ( square root of 10+2 square root of 8)( square root of 10-2 square root of 8)

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