Simplify the Radical Expression (3 square root of 2- square root of 3)(3 square root of 2+ square root of 3)

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

Step 1

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 2

Simplify and combine like terms.

Simplify each term.

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

Multiply $3$ by $3$ .

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

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

$9 ({\sqrt{2}}^{1} \sqrt{2}) + 3 \sqrt{2} \sqrt{3} - \sqrt{3} (3 \sqrt{2}) - \sqrt{3} \sqrt{3}$

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

$9 ({\sqrt{2}}^{1} {\sqrt{2}}^{1}) + 3 \sqrt{2} \sqrt{3} - \sqrt{3} (3 \sqrt{2}) - \sqrt{3} \sqrt{3}$

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

$9 {\sqrt{2}}^{1 + 1} + 3 \sqrt{2} \sqrt{3} - \sqrt{3} (3 \sqrt{2}) - \sqrt{3} \sqrt{3}$

Add $1$ and $1$ .

$9 {\sqrt{2}}^{2} + 3 \sqrt{2} \sqrt{3} - \sqrt{3} (3 \sqrt{2}) - \sqrt{3} \sqrt{3}$

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

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

$9 {(2^{\frac{1}{2}})}^{2} + 3 \sqrt{2} \sqrt{3} - \sqrt{3} (3 \sqrt{2}) - \sqrt{3} \sqrt{3}$

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

$9 \cdot 2^{\frac{1}{2} \cdot 2} + 3 \sqrt{2} \sqrt{3} - \sqrt{3} (3 \sqrt{2}) - \sqrt{3} \sqrt{3}$

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

$9 \cdot 2^{\frac{2}{2}} + 3 \sqrt{2} \sqrt{3} - \sqrt{3} (3 \sqrt{2}) - \sqrt{3} \sqrt{3}$

Cancel the common factor of $2$ .

Cancel the common factor.

$9 \cdot 2^{\frac{2}{2}} + 3 \sqrt{2} \sqrt{3} - \sqrt{3} (3 \sqrt{2}) - \sqrt{3} \sqrt{3}$

Rewrite the expression.

$9 \cdot 2^{1} + 3 \sqrt{2} \sqrt{3} - \sqrt{3} (3 \sqrt{2}) - \sqrt{3} \sqrt{3}$

Evaluate the exponent.

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

Multiply $9$ by $2$ .

$18 + 3 \sqrt{2} \sqrt{3} - \sqrt{3} (3 \sqrt{2}) - \sqrt{3} \sqrt{3}$

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

Combine using the product rule for radicals.

$18 + 3 \sqrt{3 \cdot 2} - \sqrt{3} (3 \sqrt{2}) - \sqrt{3} \sqrt{3}$

Multiply $3$ by $2$ .

$18 + 3 \sqrt{6} - \sqrt{3} (3 \sqrt{2}) - \sqrt{3} \sqrt{3}$

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

Multiply $3$ by $- 1$ .

$18 + 3 \sqrt{6} - 3 \sqrt{3} \sqrt{2} - \sqrt{3} \sqrt{3}$

Combine using the product rule for radicals.

$18 + 3 \sqrt{6} - 3 \sqrt{2 \cdot 3} - \sqrt{3} \sqrt{3}$

Multiply $2$ by $3$ .

$18 + 3 \sqrt{6} - 3 \sqrt{6} - \sqrt{3} \sqrt{3}$

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

Raise $\sqrt{3}$ to the power of $1$ .

$18 + 3 \sqrt{6} - 3 \sqrt{6} - ({\sqrt{3}}^{1} \sqrt{3})$

Raise $\sqrt{3}$ to the power of $1$ .

$18 + 3 \sqrt{6} - 3 \sqrt{6} - ({\sqrt{3}}^{1} {\sqrt{3}}^{1})$

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

$18 + 3 \sqrt{6} - 3 \sqrt{6} - {\sqrt{3}}^{1 + 1}$

Add $1$ and $1$ .

$18 + 3 \sqrt{6} - 3 \sqrt{6} - {\sqrt{3}}^{2}$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$18 + 3 \sqrt{6} - 3 \sqrt{6} - {(3^{\frac{1}{2}})}^{2}$

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

$18 + 3 \sqrt{6} - 3 \sqrt{6} - 3^{\frac{1}{2} \cdot 2}$

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

$18 + 3 \sqrt{6} - 3 \sqrt{6} - 3^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$18 + 3 \sqrt{6} - 3 \sqrt{6} - 3^{\frac{2}{2}}$

Rewrite the expression.

$18 + 3 \sqrt{6} - 3 \sqrt{6} - 3^{1}$

Evaluate the exponent.

$18 + 3 \sqrt{6} - 3 \sqrt{6} - 1 \cdot 3$

Multiply $- 1$ by $3$ .

$18 + 3 \sqrt{6} - 3 \sqrt{6} - 3$

Subtract $3$ from $18$ .

$15 + 3 \sqrt{6} - 3 \sqrt{6}$

Subtract $3 \sqrt{6}$ from $3 \sqrt{6}$ .

$15 + 0$

Add $15$ and $0$ .

$15$

Simplify the Radical Expression (3 square root of 2- square root of 3)(3 square root of 2+ square root of 3)

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