Simplify ( square root of 10+ square root of 5)( square root of 10- square root of 5)

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

Step 1

Expand $(\sqrt{10} + \sqrt{5}) (\sqrt{10} - \sqrt{5})$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 2

Simplify and combine like terms.

Simplify each term.

Combine using the product rule for radicals.

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

Multiply $10$ by $10$ .

$\sqrt{100} + \sqrt{10} (- \sqrt{5}) + \sqrt{5} \sqrt{10} + \sqrt{5} (- \sqrt{5})$

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

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

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

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

Multiply $\sqrt{10} (- \sqrt{5})$ .

Combine using the product rule for radicals.

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

Multiply $10$ by $5$ .

$10 - \sqrt{50} + \sqrt{5} \sqrt{10} + \sqrt{5} (- \sqrt{5})$

Rewrite $50$ as $5^{2} \cdot 2$ .

Factor $25$ out of $50$ .

$10 - \sqrt{25 (2)} + \sqrt{5} \sqrt{10} + \sqrt{5} (- \sqrt{5})$

Rewrite $25$ as $5^{2}$ .

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

Pull terms out from under the radical.

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

Multiply $5$ by $- 1$ .

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

Combine using the product rule for radicals.

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

Multiply $5$ by $10$ .

$10 - 5 \sqrt{2} + \sqrt{50} + \sqrt{5} (- \sqrt{5})$

Rewrite $50$ as $5^{2} \cdot 2$ .

Factor $25$ out of $50$ .

$10 - 5 \sqrt{2} + \sqrt{25 (2)} + \sqrt{5} (- \sqrt{5})$

Rewrite $25$ as $5^{2}$ .

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

Pull terms out from under the radical.

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

Multiply $\sqrt{5} (- \sqrt{5})$ .

Raise $\sqrt{5}$ to the power of $1$ .

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

Raise $\sqrt{5}$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Multiply $- 1$ by $5$ .

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

Subtract $5$ from $10$ .

$5 - 5 \sqrt{2} + 5 \sqrt{2}$

Add $- 5 \sqrt{2}$ and $5 \sqrt{2}$ .

$5 + 0$

Add $5$ and $0$ .

$5$

Simplify ( square root of 10+ square root of 5)( square root of 10- square root of 5)

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