Simplify ( square root of 3+ square root of 10)( square root of 3- square root of 10)

$(\sqrt{3} + \sqrt{10}) (\sqrt{3} - \sqrt{10})$

Step 1

Expand $(\sqrt{3} + \sqrt{10}) (\sqrt{3} - \sqrt{10})$ using the FOIL Method.

Apply the distributive property.

$\sqrt{3} (\sqrt{3} - \sqrt{10}) + \sqrt{10} (\sqrt{3} - \sqrt{10})$

Apply the distributive property.

$\sqrt{3} \sqrt{3} + \sqrt{3} (- \sqrt{10}) + \sqrt{10} (\sqrt{3} - \sqrt{10})$

Apply the distributive property.

$\sqrt{3} \sqrt{3} + \sqrt{3} (- \sqrt{10}) + \sqrt{10} \sqrt{3} + \sqrt{10} (- \sqrt{10})$

Step 2

Simplify and combine like terms.

Simplify each term.

Combine using the product rule for radicals.

$\sqrt{3 \cdot 3} + \sqrt{3} (- \sqrt{10}) + \sqrt{10} \sqrt{3} + \sqrt{10} (- \sqrt{10})$

Multiply $3$ by $3$ .

$\sqrt{9} + \sqrt{3} (- \sqrt{10}) + \sqrt{10} \sqrt{3} + \sqrt{10} (- \sqrt{10})$

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

$\sqrt{3^{2}} + \sqrt{3} (- \sqrt{10}) + \sqrt{10} \sqrt{3} + \sqrt{10} (- \sqrt{10})$

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

$3 + \sqrt{3} (- \sqrt{10}) + \sqrt{10} \sqrt{3} + \sqrt{10} (- \sqrt{10})$

Multiply $\sqrt{3} (- \sqrt{10})$ .

Combine using the product rule for radicals.

$3 - \sqrt{3 \cdot 10} + \sqrt{10} \sqrt{3} + \sqrt{10} (- \sqrt{10})$

Multiply $3$ by $10$ .

$3 - \sqrt{30} + \sqrt{10} \sqrt{3} + \sqrt{10} (- \sqrt{10})$

Combine using the product rule for radicals.

$3 - \sqrt{30} + \sqrt{10 \cdot 3} + \sqrt{10} (- \sqrt{10})$

Multiply $10$ by $3$ .

$3 - \sqrt{30} + \sqrt{30} + \sqrt{10} (- \sqrt{10})$

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

Raise $\sqrt{10}$ to the power of $1$ .

$3 - \sqrt{30} + \sqrt{30} - ({\sqrt{10}}^{1} \sqrt{10})$

Raise $\sqrt{10}$ to the power of $1$ .

$3 - \sqrt{30} + \sqrt{30} - ({\sqrt{10}}^{1} {\sqrt{10}}^{1})$

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

$3 - \sqrt{30} + \sqrt{30} - {\sqrt{10}}^{1 + 1}$

Add $1$ and $1$ .

$3 - \sqrt{30} + \sqrt{30} - {\sqrt{10}}^{2}$

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$3 - \sqrt{30} + \sqrt{30} - {(10^{\frac{1}{2}})}^{2}$

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

$3 - \sqrt{30} + \sqrt{30} - 10^{\frac{1}{2} \cdot 2}$

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

$3 - \sqrt{30} + \sqrt{30} - 10^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$3 - \sqrt{30} + \sqrt{30} - 10^{\frac{2}{2}}$

Rewrite the expression.

$3 - \sqrt{30} + \sqrt{30} - 10^{1}$

Evaluate the exponent.

$3 - \sqrt{30} + \sqrt{30} - 1 \cdot 10$

Multiply $- 1$ by $10$ .

$3 - \sqrt{30} + \sqrt{30} - 10$

Subtract $10$ from $3$ .

$- 7 - \sqrt{30} + \sqrt{30}$

Add $- \sqrt{30}$ and $\sqrt{30}$ .

$- 7 + 0$

Add $- 7$ and $0$ .

$- 7$

Simplify ( square root of 3+ square root of 10)( square root of 3- square root of 10)

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