Simplify (3 square root of 2-2 square root of 3)/(3 square root of 2+2 square root of 3)

$\frac{3 \sqrt{2} - 2 \sqrt{3}}{3 \sqrt{2} + 2 \sqrt{3}}$

Step 1

Multiply $\frac{3 \sqrt{2} - 2 \sqrt{3}}{3 \sqrt{2} + 2 \sqrt{3}}$ by $\frac{3 \sqrt{2} - 2 \sqrt{3}}{3 \sqrt{2} - 2 \sqrt{3}}$ .

$\frac{3 \sqrt{2} - 2 \sqrt{3}}{3 \sqrt{2} + 2 \sqrt{3}} \cdot \frac{3 \sqrt{2} - 2 \sqrt{3}}{3 \sqrt{2} - 2 \sqrt{3}}$

Step 2

Combine fractions.

Multiply $\frac{3 \sqrt{2} - 2 \sqrt{3}}{3 \sqrt{2} + 2 \sqrt{3}}$ by $\frac{3 \sqrt{2} - 2 \sqrt{3}}{3 \sqrt{2} - 2 \sqrt{3}}$ .

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

Expand the denominator using the FOIL method.

$\frac{(3 \sqrt{2} - 2 \sqrt{3}) (3 \sqrt{2} - 2 \sqrt{3})}{9 {\sqrt{2}}^{2} - 6 \sqrt{6} + 6 \sqrt{6} - 4 {\sqrt{3}}^{2}}$

Simplify.

$\frac{(3 \sqrt{2} - 2 \sqrt{3}) (3 \sqrt{2} - 2 \sqrt{3})}{6}$

Step 3

Simplify the numerator.

Raise $3 \sqrt{2} - 2 \sqrt{3}$ to the power of $1$ .

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

Raise $3 \sqrt{2} - 2 \sqrt{3}$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

Step 4

Rewrite ${(3 \sqrt{2} - 2 \sqrt{3})}^{2}$ as $(3 \sqrt{2} - 2 \sqrt{3}) (3 \sqrt{2} - 2 \sqrt{3})$ .

$\frac{(3 \sqrt{2} - 2 \sqrt{3}) (3 \sqrt{2} - 2 \sqrt{3})}{6}$

Step 5

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

Apply the distributive property.

$\frac{3 \sqrt{2} (3 \sqrt{2} - 2 \sqrt{3}) - 2 \sqrt{3} (3 \sqrt{2} - 2 \sqrt{3})}{6}$

Apply the distributive property.

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

Apply the distributive property.

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

Step 6

Simplify and combine like terms.

Simplify each term.

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

Multiply $3$ by $3$ .

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

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

$\frac{9 ({\sqrt{2}}^{1} \sqrt{2}) + 3 \sqrt{2} (- 2 \sqrt{3}) - 2 \sqrt{3} (3 \sqrt{2}) - 2 \sqrt{3} (- 2 \sqrt{3})}{6}$

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

$\frac{9 ({\sqrt{2}}^{1} {\sqrt{2}}^{1}) + 3 \sqrt{2} (- 2 \sqrt{3}) - 2 \sqrt{3} (3 \sqrt{2}) - 2 \sqrt{3} (- 2 \sqrt{3})}{6}$

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

$\frac{9 {\sqrt{2}}^{1 + 1} + 3 \sqrt{2} (- 2 \sqrt{3}) - 2 \sqrt{3} (3 \sqrt{2}) - 2 \sqrt{3} (- 2 \sqrt{3})}{6}$

Add $1$ and $1$ .

$\frac{9 {\sqrt{2}}^{2} + 3 \sqrt{2} (- 2 \sqrt{3}) - 2 \sqrt{3} (3 \sqrt{2}) - 2 \sqrt{3} (- 2 \sqrt{3})}{6}$

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

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

$\frac{9 {(2^{\frac{1}{2}})}^{2} + 3 \sqrt{2} (- 2 \sqrt{3}) - 2 \sqrt{3} (3 \sqrt{2}) - 2 \sqrt{3} (- 2 \sqrt{3})}{6}$

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

$\frac{9 \cdot 2^{\frac{1}{2} \cdot 2} + 3 \sqrt{2} (- 2 \sqrt{3}) - 2 \sqrt{3} (3 \sqrt{2}) - 2 \sqrt{3} (- 2 \sqrt{3})}{6}$

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

$\frac{9 \cdot 2^{\frac{2}{2}} + 3 \sqrt{2} (- 2 \sqrt{3}) - 2 \sqrt{3} (3 \sqrt{2}) - 2 \sqrt{3} (- 2 \sqrt{3})}{6}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{9 \cdot 2^{\frac{2}{2}} + 3 \sqrt{2} (- 2 \sqrt{3}) - 2 \sqrt{3} (3 \sqrt{2}) - 2 \sqrt{3} (- 2 \sqrt{3})}{6}$

Rewrite the expression.

$\frac{9 \cdot 2^{1} + 3 \sqrt{2} (- 2 \sqrt{3}) - 2 \sqrt{3} (3 \sqrt{2}) - 2 \sqrt{3} (- 2 \sqrt{3})}{6}$

Evaluate the exponent.

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

Multiply $9$ by $2$ .

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

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

Multiply $- 2$ by $3$ .

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

Combine using the product rule for radicals.

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

Multiply $3$ by $2$ .

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

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

Multiply $3$ by $- 2$ .

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

Combine using the product rule for radicals.

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

Multiply $2$ by $3$ .

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

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

Multiply $- 2$ by $- 2$ .

$\frac{18 - 6 \sqrt{6} - 6 \sqrt{6} + 4 \sqrt{3} \sqrt{3}}{6}$

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

$\frac{18 - 6 \sqrt{6} - 6 \sqrt{6} + 4 ({\sqrt{3}}^{1} \sqrt{3})}{6}$

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

$\frac{18 - 6 \sqrt{6} - 6 \sqrt{6} + 4 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}{6}$

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

$\frac{18 - 6 \sqrt{6} - 6 \sqrt{6} + 4 {\sqrt{3}}^{1 + 1}}{6}$

Add $1$ and $1$ .

$\frac{18 - 6 \sqrt{6} - 6 \sqrt{6} + 4 {\sqrt{3}}^{2}}{6}$

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

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

$\frac{18 - 6 \sqrt{6} - 6 \sqrt{6} + 4 {(3^{\frac{1}{2}})}^{2}}{6}$

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

$\frac{18 - 6 \sqrt{6} - 6 \sqrt{6} + 4 \cdot 3^{\frac{1}{2} \cdot 2}}{6}$

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

$\frac{18 - 6 \sqrt{6} - 6 \sqrt{6} + 4 \cdot 3^{\frac{2}{2}}}{6}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{18 - 6 \sqrt{6} - 6 \sqrt{6} + 4 \cdot 3^{\frac{2}{2}}}{6}$

Rewrite the expression.

$\frac{18 - 6 \sqrt{6} - 6 \sqrt{6} + 4 \cdot 3^{1}}{6}$

Evaluate the exponent.

$\frac{18 - 6 \sqrt{6} - 6 \sqrt{6} + 4 \cdot 3}{6}$

Multiply $4$ by $3$ .

$\frac{18 - 6 \sqrt{6} - 6 \sqrt{6} + 12}{6}$

Add $18$ and $12$ .

$\frac{30 - 6 \sqrt{6} - 6 \sqrt{6}}{6}$

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

$\frac{30 - 12 \sqrt{6}}{6}$

Step 7

Cancel the common factor of $30 - 12 \sqrt{6}$ and $6$ .

Factor $6$ out of $30$ .

$\frac{6 \cdot 5 - 12 \sqrt{6}}{6}$

Factor $6$ out of $- 12 \sqrt{6}$ .

$\frac{6 \cdot 5 + 6 (- 2 \sqrt{6})}{6}$

Factor $6$ out of $6 (5) + 6 (- 2 \sqrt{6})$ .

$\frac{6 (5 - 2 \sqrt{6})}{6}$

Cancel the common factors.

Factor $6$ out of $6$ .

$\frac{6 (5 - 2 \sqrt{6})}{6 (1)}$

Cancel the common factor.

$\frac{6 (5 - 2 \sqrt{6})}{6 \cdot 1}$

Rewrite the expression.

$\frac{5 - 2 \sqrt{6}}{1}$

Divide $5 - 2 \sqrt{6}$ by $1$ .

$5 - 2 \sqrt{6}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$5 - 2 \sqrt{6}$

Decimal Form:

$0.10102051 \dots$

Simplify (3 square root of 2-2 square root of 3)/(3 square root of 2+2 square root of 3)

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