Simplify (3 square root of 2+ square root of 5)( square root of 2-3 square root of 5r)

$(3 \sqrt{2} + \sqrt{5}) (\sqrt{2} - 3 \sqrt{5 r})$

Expand $(3 \sqrt{2} + \sqrt{5}) (\sqrt{2} - 3 \sqrt{5 r})$ using the FOIL Method.

Apply the distributive property.

$3 \sqrt{2} (\sqrt{2} - 3 \sqrt{5 r}) + \sqrt{5} (\sqrt{2} - 3 \sqrt{5 r})$

Apply the distributive property.

$3 \sqrt{2} \sqrt{2} + 3 \sqrt{2} (- 3 \sqrt{5 r}) + \sqrt{5} (\sqrt{2} - 3 \sqrt{5 r})$

Apply the distributive property.

$3 \sqrt{2} \sqrt{2} + 3 \sqrt{2} (- 3 \sqrt{5 r}) + \sqrt{5} \sqrt{2} + \sqrt{5} (- 3 \sqrt{5 r})$

Simplify each term.

Multiply $3 \sqrt{2} \sqrt{2}$ .

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

$3 ({\sqrt{2}}^{1} \sqrt{2}) + 3 \sqrt{2} (- 3 \sqrt{5 r}) + \sqrt{5} \sqrt{2} + \sqrt{5} (- 3 \sqrt{5 r})$

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

$3 ({\sqrt{2}}^{1} {\sqrt{2}}^{1}) + 3 \sqrt{2} (- 3 \sqrt{5 r}) + \sqrt{5} \sqrt{2} + \sqrt{5} (- 3 \sqrt{5 r})$

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

$3 {\sqrt{2}}^{1 + 1} + 3 \sqrt{2} (- 3 \sqrt{5 r}) + \sqrt{5} \sqrt{2} + \sqrt{5} (- 3 \sqrt{5 r})$

Add $1$ and $1$ .

$3 {\sqrt{2}}^{2} + 3 \sqrt{2} (- 3 \sqrt{5 r}) + \sqrt{5} \sqrt{2} + \sqrt{5} (- 3 \sqrt{5 r})$

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

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

$3 {(2^{\frac{1}{2}})}^{2} + 3 \sqrt{2} (- 3 \sqrt{5 r}) + \sqrt{5} \sqrt{2} + \sqrt{5} (- 3 \sqrt{5 r})$

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

$3 \cdot 2^{\frac{1}{2} \cdot 2} + 3 \sqrt{2} (- 3 \sqrt{5 r}) + \sqrt{5} \sqrt{2} + \sqrt{5} (- 3 \sqrt{5 r})$

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

$3 \cdot 2^{\frac{2}{2}} + 3 \sqrt{2} (- 3 \sqrt{5 r}) + \sqrt{5} \sqrt{2} + \sqrt{5} (- 3 \sqrt{5 r})$

Cancel the common factor of $2$ .

Cancel the common factor.

$3 \cdot 2^{\frac{2}{2}} + 3 \sqrt{2} (- 3 \sqrt{5 r}) + \sqrt{5} \sqrt{2} + \sqrt{5} (- 3 \sqrt{5 r})$

Rewrite the expression.

$3 \cdot 2^{1} + 3 \sqrt{2} (- 3 \sqrt{5 r}) + \sqrt{5} \sqrt{2} + \sqrt{5} (- 3 \sqrt{5 r})$

Evaluate the exponent.

$3 \cdot 2 + 3 \sqrt{2} (- 3 \sqrt{5 r}) + \sqrt{5} \sqrt{2} + \sqrt{5} (- 3 \sqrt{5 r})$

Multiply $3$ by $2$ .

$6 + 3 \sqrt{2} (- 3 \sqrt{5 r}) + \sqrt{5} \sqrt{2} + \sqrt{5} (- 3 \sqrt{5 r})$

Multiply $3 \sqrt{2} (- 3 \sqrt{5 r})$ .

Multiply $- 3$ by $3$ .

$6 - 9 \sqrt{2} \sqrt{5 r} + \sqrt{5} \sqrt{2} + \sqrt{5} (- 3 \sqrt{5 r})$

Combine using the product rule for radicals.

$6 - 9 \sqrt{5 r \cdot 2} + \sqrt{5} \sqrt{2} + \sqrt{5} (- 3 \sqrt{5 r})$

Multiply $2$ by $5$ .

$6 - 9 \sqrt{10 r} + \sqrt{5} \sqrt{2} + \sqrt{5} (- 3 \sqrt{5 r})$

Combine using the product rule for radicals.

$6 - 9 \sqrt{10 r} + \sqrt{5 \cdot 2} + \sqrt{5} (- 3 \sqrt{5 r})$

Multiply $5$ by $2$ .

$6 - 9 \sqrt{10 r} + \sqrt{10} + \sqrt{5} (- 3 \sqrt{5 r})$

Multiply $\sqrt{5} (- 3 \sqrt{5 r})$ .

Combine using the product rule for radicals.

$6 - 9 \sqrt{10 r} + \sqrt{10} - 3 \sqrt{5 (5 r)}$

Multiply $5$ by $5$ .

$6 - 9 \sqrt{10 r} + \sqrt{10} - 3 \sqrt{25 r}$

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

$6 - 9 \sqrt{10 r} + \sqrt{10} - 3 \sqrt{5^{2} r}$

Pull terms out from under the radical.

$6 - 9 \sqrt{10 r} + \sqrt{10} - 3 (5 \sqrt{r})$

Multiply $5$ by $- 3$ .

$6 - 9 \sqrt{10 r} + \sqrt{10} - 15 \sqrt{r}$

Simplify (3 square root of 2+ square root of 5)( square root of 2-3 square root of 5r)

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