Evaluate (2 square root of 3+5 square root of 2)/(3 square root of 3-2 square root of 2)

$\frac{2 \sqrt{3} + 5 \sqrt{2}}{3 \sqrt{3} - 2 \sqrt{2}}$

Step 1

Multiply $\frac{2 \sqrt{3} + 5 \sqrt{2}}{3 \sqrt{3} - 2 \sqrt{2}}$ by $\frac{3 \sqrt{3} + 2 \sqrt{2}}{3 \sqrt{3} + 2 \sqrt{2}}$ .

$\frac{2 \sqrt{3} + 5 \sqrt{2}}{3 \sqrt{3} - 2 \sqrt{2}} \cdot \frac{3 \sqrt{3} + 2 \sqrt{2}}{3 \sqrt{3} + 2 \sqrt{2}}$

Step 2

Combine fractions.

Multiply $\frac{2 \sqrt{3} + 5 \sqrt{2}}{3 \sqrt{3} - 2 \sqrt{2}}$ by $\frac{3 \sqrt{3} + 2 \sqrt{2}}{3 \sqrt{3} + 2 \sqrt{2}}$ .

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

Expand the denominator using the FOIL method.

$\frac{(2 \sqrt{3} + 5 \sqrt{2}) (3 \sqrt{3} + 2 \sqrt{2})}{9 {\sqrt{3}}^{2} + 6 \sqrt{6} - 6 \sqrt{6} - 4 {\sqrt{2}}^{2}}$

Simplify.

$\frac{(2 \sqrt{3} + 5 \sqrt{2}) (3 \sqrt{3} + 2 \sqrt{2})}{19}$

Step 3

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

Apply the distributive property.

$\frac{2 \sqrt{3} (3 \sqrt{3} + 2 \sqrt{2}) + 5 \sqrt{2} (3 \sqrt{3} + 2 \sqrt{2})}{19}$

Apply the distributive property.

$\frac{2 \sqrt{3} (3 \sqrt{3}) + 2 \sqrt{3} (2 \sqrt{2}) + 5 \sqrt{2} (3 \sqrt{3} + 2 \sqrt{2})}{19}$

Apply the distributive property.

$\frac{2 \sqrt{3} (3 \sqrt{3}) + 2 \sqrt{3} (2 \sqrt{2}) + 5 \sqrt{2} (3 \sqrt{3}) + 5 \sqrt{2} (2 \sqrt{2})}{19}$

Step 4

Simplify and combine like terms.

Simplify each term.

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

Multiply $3$ by $2$ .

$\frac{6 \sqrt{3} \sqrt{3} + 2 \sqrt{3} (2 \sqrt{2}) + 5 \sqrt{2} (3 \sqrt{3}) + 5 \sqrt{2} (2 \sqrt{2})}{19}$

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

$\frac{6 ({\sqrt{3}}^{1} \sqrt{3}) + 2 \sqrt{3} (2 \sqrt{2}) + 5 \sqrt{2} (3 \sqrt{3}) + 5 \sqrt{2} (2 \sqrt{2})}{19}$

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

$\frac{6 ({\sqrt{3}}^{1} {\sqrt{3}}^{1}) + 2 \sqrt{3} (2 \sqrt{2}) + 5 \sqrt{2} (3 \sqrt{3}) + 5 \sqrt{2} (2 \sqrt{2})}{19}$

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

$\frac{6 {\sqrt{3}}^{1 + 1} + 2 \sqrt{3} (2 \sqrt{2}) + 5 \sqrt{2} (3 \sqrt{3}) + 5 \sqrt{2} (2 \sqrt{2})}{19}$

Add $1$ and $1$ .

$\frac{6 {\sqrt{3}}^{2} + 2 \sqrt{3} (2 \sqrt{2}) + 5 \sqrt{2} (3 \sqrt{3}) + 5 \sqrt{2} (2 \sqrt{2})}{19}$

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

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

$\frac{6 {(3^{\frac{1}{2}})}^{2} + 2 \sqrt{3} (2 \sqrt{2}) + 5 \sqrt{2} (3 \sqrt{3}) + 5 \sqrt{2} (2 \sqrt{2})}{19}$

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

$\frac{6 \cdot 3^{\frac{1}{2} \cdot 2} + 2 \sqrt{3} (2 \sqrt{2}) + 5 \sqrt{2} (3 \sqrt{3}) + 5 \sqrt{2} (2 \sqrt{2})}{19}$

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

$\frac{6 \cdot 3^{\frac{2}{2}} + 2 \sqrt{3} (2 \sqrt{2}) + 5 \sqrt{2} (3 \sqrt{3}) + 5 \sqrt{2} (2 \sqrt{2})}{19}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{6 \cdot 3^{\frac{2}{2}} + 2 \sqrt{3} (2 \sqrt{2}) + 5 \sqrt{2} (3 \sqrt{3}) + 5 \sqrt{2} (2 \sqrt{2})}{19}$

Rewrite the expression.

$\frac{6 \cdot 3^{1} + 2 \sqrt{3} (2 \sqrt{2}) + 5 \sqrt{2} (3 \sqrt{3}) + 5 \sqrt{2} (2 \sqrt{2})}{19}$

Evaluate the exponent.

$\frac{6 \cdot 3 + 2 \sqrt{3} (2 \sqrt{2}) + 5 \sqrt{2} (3 \sqrt{3}) + 5 \sqrt{2} (2 \sqrt{2})}{19}$

Multiply $6$ by $3$ .

$\frac{18 + 2 \sqrt{3} (2 \sqrt{2}) + 5 \sqrt{2} (3 \sqrt{3}) + 5 \sqrt{2} (2 \sqrt{2})}{19}$

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

Multiply $2$ by $2$ .

$\frac{18 + 4 \sqrt{3} \sqrt{2} + 5 \sqrt{2} (3 \sqrt{3}) + 5 \sqrt{2} (2 \sqrt{2})}{19}$

Combine using the product rule for radicals.

$\frac{18 + 4 \sqrt{2 \cdot 3} + 5 \sqrt{2} (3 \sqrt{3}) + 5 \sqrt{2} (2 \sqrt{2})}{19}$

Multiply $2$ by $3$ .

$\frac{18 + 4 \sqrt{6} + 5 \sqrt{2} (3 \sqrt{3}) + 5 \sqrt{2} (2 \sqrt{2})}{19}$

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

Multiply $3$ by $5$ .

$\frac{18 + 4 \sqrt{6} + 15 \sqrt{2} \sqrt{3} + 5 \sqrt{2} (2 \sqrt{2})}{19}$

Combine using the product rule for radicals.

$\frac{18 + 4 \sqrt{6} + 15 \sqrt{3 \cdot 2} + 5 \sqrt{2} (2 \sqrt{2})}{19}$

Multiply $3$ by $2$ .

$\frac{18 + 4 \sqrt{6} + 15 \sqrt{6} + 5 \sqrt{2} (2 \sqrt{2})}{19}$

Multiply $5 \sqrt{2} (2 \sqrt{2})$ .

Multiply $2$ by $5$ .

$\frac{18 + 4 \sqrt{6} + 15 \sqrt{6} + 10 \sqrt{2} \sqrt{2}}{19}$

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

$\frac{18 + 4 \sqrt{6} + 15 \sqrt{6} + 10 ({\sqrt{2}}^{1} \sqrt{2})}{19}$

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

$\frac{18 + 4 \sqrt{6} + 15 \sqrt{6} + 10 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}{19}$

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

$\frac{18 + 4 \sqrt{6} + 15 \sqrt{6} + 10 {\sqrt{2}}^{1 + 1}}{19}$

Add $1$ and $1$ .

$\frac{18 + 4 \sqrt{6} + 15 \sqrt{6} + 10 {\sqrt{2}}^{2}}{19}$

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

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

$\frac{18 + 4 \sqrt{6} + 15 \sqrt{6} + 10 {(2^{\frac{1}{2}})}^{2}}{19}$

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

$\frac{18 + 4 \sqrt{6} + 15 \sqrt{6} + 10 \cdot 2^{\frac{1}{2} \cdot 2}}{19}$

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

$\frac{18 + 4 \sqrt{6} + 15 \sqrt{6} + 10 \cdot 2^{\frac{2}{2}}}{19}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{18 + 4 \sqrt{6} + 15 \sqrt{6} + 10 \cdot 2^{\frac{2}{2}}}{19}$

Rewrite the expression.

$\frac{18 + 4 \sqrt{6} + 15 \sqrt{6} + 10 \cdot 2^{1}}{19}$

Evaluate the exponent.

$\frac{18 + 4 \sqrt{6} + 15 \sqrt{6} + 10 \cdot 2}{19}$

Multiply $10$ by $2$ .

$\frac{18 + 4 \sqrt{6} + 15 \sqrt{6} + 20}{19}$

Add $18$ and $20$ .

$\frac{38 + 4 \sqrt{6} + 15 \sqrt{6}}{19}$

Add $4 \sqrt{6}$ and $15 \sqrt{6}$ .

$\frac{38 + 19 \sqrt{6}}{19}$

Step 5

Cancel the common factor of $38 + 19 \sqrt{6}$ and $19$ .

Factor $19$ out of $38$ .

$\frac{19 \cdot 2 + 19 \sqrt{6}}{19}$

Factor $19$ out of $19 \sqrt{6}$ .

$\frac{19 \cdot 2 + 19 (\sqrt{6})}{19}$

Factor $19$ out of $19 (2) + 19 (\sqrt{6})$ .

$\frac{19 (2 + \sqrt{6})}{19}$

Cancel the common factors.

Factor $19$ out of $19$ .

$\frac{19 (2 + \sqrt{6})}{19 (1)}$

Cancel the common factor.

$\frac{19 (2 + \sqrt{6})}{19 \cdot 1}$

Rewrite the expression.

$\frac{2 + \sqrt{6}}{1}$

Divide $2 + \sqrt{6}$ by $1$ .

$2 + \sqrt{6}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$2 + \sqrt{6}$

Decimal Form:

$4.44948974 \dots$

Evaluate (2 square root of 3+5 square root of 2)/(3 square root of 3-2 square root of 2)

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