$\frac{3\sqrt{32} + 2\sqrt{125}}{\sqrt{2} - 3\sqrt{5}}$
Factor $16$$\frac{3\sqrt{16\left(2\right)} + 2\sqrt{125}}{\sqrt{2} - 3\sqrt{5}}$
Rewrite $16$$\frac{3\sqrt{4^{2} \cdot 2} + 2\sqrt{125}}{\sqrt{2} - 3\sqrt{5}}$
Pull terms out from under the radical$\frac{3\left(4\sqrt{2}\right) + 2\sqrt{125}}{\sqrt{2} - 3\sqrt{5}}$
Multiply $4$$\frac{12\sqrt{2} + 2\sqrt{125}}{\sqrt{2} - 3\sqrt{5}}$
Factor $25$$\frac{12\sqrt{2} + 2\sqrt{25\left(5\right)}}{\sqrt{2} - 3\sqrt{5}}$
Rewrite $25$$\frac{12\sqrt{2} + 2\sqrt{5^{2} \cdot 5}}{\sqrt{2} - 3\sqrt{5}}$
Pull terms out from under the radical$\frac{12\sqrt{2} + 2\left(5\sqrt{5}\right)}{\sqrt{2} - 3\sqrt{5}}$
Multiply $5$$\frac{12\sqrt{2} + 10\sqrt{5}}{\sqrt{2} - 3\sqrt{5}}$
Multiply $\frac{12\sqrt{2} + 10\sqrt{5}}{\sqrt{2} - 3\sqrt{5}}$$\frac{12\sqrt{2} + 10\sqrt{5}}{\sqrt{2} - 3\sqrt{5}} \cdot \frac{\sqrt{2} + 3\sqrt{5}}{\sqrt{2} + 3\sqrt{5}}$
Multiply $\frac{12\sqrt{2} + 10\sqrt{5}}{\sqrt{2} - 3\sqrt{5}}$$\frac{\left(12\sqrt{2} + 10\sqrt{5}\right)\left(\sqrt{2} + 3\sqrt{5}\right)}{\left(\sqrt{2} - 3\sqrt{5}\right)\left(\sqrt{2} + 3\sqrt{5}\right)}$
Expand the denominator using the foil method$\frac{\left(12\sqrt{2} + 10\sqrt{5}\right)\left(\sqrt{2} + 3\sqrt{5}\right)}{\sqrt{2}^{2} + 3\sqrt{10} - 3\sqrt{10} - 9\sqrt{5}^{2}}$
Simplify$\frac{\left(12\sqrt{2} + 10\sqrt{5}\right)\left(\sqrt{2} + 3\sqrt{5}\right)}{ - 43}$
Apply the distributive property$\frac{12\sqrt{2}\left(\sqrt{2} + 3\sqrt{5}\right) + 10\sqrt{5}\left(\sqrt{2} + 3\sqrt{5}\right)}{ - 43}$
Apply the distributive property$\frac{12\sqrt{2}\sqrt{2} + 12\sqrt{2}\left(3\sqrt{5}\right) + 10\sqrt{5}\left(\sqrt{2} + 3\sqrt{5}\right)}{ - 43}$
Apply the distributive property$\frac{12\sqrt{2}\sqrt{2} + 12\sqrt{2}\left(3\sqrt{5}\right) + 10\sqrt{5}\sqrt{2} + 10\sqrt{5}\left(3\sqrt{5}\right)}{ - 43}$
Raise $\sqrt{2}$$\frac{12\left(\sqrt{2}^{1}\sqrt{2}\right) + 12\sqrt{2}\left(3\sqrt{5}\right) + 10\sqrt{5}\sqrt{2} + 10\sqrt{5}\left(3\sqrt{5}\right)}{ - 43}$
Raise $\sqrt{2}$$\frac{12\left(\sqrt{2}^{1}\sqrt{2}^{1}\right) + 12\sqrt{2}\left(3\sqrt{5}\right) + 10\sqrt{5}\sqrt{2} + 10\sqrt{5}\left(3\sqrt{5}\right)}{ - 43}$
Use the power rule $a^{m}a^{n} = a^{m + n}$$\frac{12\sqrt{2}^{1 + 1} + 12\sqrt{2}\left(3\sqrt{5}\right) + 10\sqrt{5}\sqrt{2} + 10\sqrt{5}\left(3\sqrt{5}\right)}{ - 43}$
Add $1$$\frac{12\sqrt{2}^{2} + 12\sqrt{2}\left(3\sqrt{5}\right) + 10\sqrt{5}\sqrt{2} + 10\sqrt{5}\left(3\sqrt{5}\right)}{ - 43}$
Use $a^{x}n = a^{\frac{x}{n}}$$\frac{12\left(2^{\frac{1}{2}}\right)^{2} + 12\sqrt{2}\left(3\sqrt{5}\right) + 10\sqrt{5}\sqrt{2} + 10\sqrt{5}\left(3\sqrt{5}\right)}{ - 43}$
Apply the power rule and multiply exponents, $\left(a^{m}\right)^{n} = a^{mn}$$\frac{12 \cdot 2^{\frac{1}{2} \cdot 2} + 12\sqrt{2}\left(3\sqrt{5}\right) + 10\sqrt{5}\sqrt{2} + 10\sqrt{5}\left(3\sqrt{5}\right)}{ - 43}$
Combine $\frac{1}{2}$$\frac{12 \cdot 2^{\frac{2}{2}} + 12\sqrt{2}\left(3\sqrt{5}\right) + 10\sqrt{5}\sqrt{2} + 10\sqrt{5}\left(3\sqrt{5}\right)}{ - 43}$
Cancel the common factor$\frac{12 \cdot 2^{\frac{2}{2}} + 12\sqrt{2}\left(3\sqrt{5}\right) + 10\sqrt{5}\sqrt{2} + 10\sqrt{5}\left(3\sqrt{5}\right)}{ - 43}$
Rewrite the expression$\frac{12 \cdot 2^{1} + 12\sqrt{2}\left(3\sqrt{5}\right) + 10\sqrt{5}\sqrt{2} + 10\sqrt{5}\left(3\sqrt{5}\right)}{ - 43}$
Evaluate the exponent$\frac{12 \cdot 2 + 12\sqrt{2}\left(3\sqrt{5}\right) + 10\sqrt{5}\sqrt{2} + 10\sqrt{5}\left(3\sqrt{5}\right)}{ - 43}$
Multiply $12$$\frac{24 + 12\sqrt{2}\left(3\sqrt{5}\right) + 10\sqrt{5}\sqrt{2} + 10\sqrt{5}\left(3\sqrt{5}\right)}{ - 43}$
Multiply $3$$\frac{24 + 36\sqrt{2}\sqrt{5} + 10\sqrt{5}\sqrt{2} + 10\sqrt{5}\left(3\sqrt{5}\right)}{ - 43}$
Combine using the product rule for radicals$\frac{24 + 36\sqrt{5 \cdot 2} + 10\sqrt{5}\sqrt{2} + 10\sqrt{5}\left(3\sqrt{5}\right)}{ - 43}$
Multiply $5$$\frac{24 + 36\sqrt{10} + 10\sqrt{5}\sqrt{2} + 10\sqrt{5}\left(3\sqrt{5}\right)}{ - 43}$
Combine using the product rule for radicals$\frac{24 + 36\sqrt{10} + 10\sqrt{2 \cdot 5} + 10\sqrt{5}\left(3\sqrt{5}\right)}{ - 43}$
Multiply $2$$\frac{24 + 36\sqrt{10} + 10\sqrt{10} + 10\sqrt{5}\left(3\sqrt{5}\right)}{ - 43}$
Multiply $3$$\frac{24 + 36\sqrt{10} + 10\sqrt{10} + 30\sqrt{5}\sqrt{5}}{ - 43}$
Raise $\sqrt{5}$$\frac{24 + 36\sqrt{10} + 10\sqrt{10} + 30\left(\sqrt{5}^{1}\sqrt{5}\right)}{ - 43}$
Raise $\sqrt{5}$$\frac{24 + 36\sqrt{10} + 10\sqrt{10} + 30\left(\sqrt{5}^{1}\sqrt{5}^{1}\right)}{ - 43}$
Use the power rule $a^{m}a^{n} = a^{m + n}$$\frac{24 + 36\sqrt{10} + 10\sqrt{10} + 30\sqrt{5}^{1 + 1}}{ - 43}$
Add $1$$\frac{24 + 36\sqrt{10} + 10\sqrt{10} + 30\sqrt{5}^{2}}{ - 43}$
Use $a^{x}n = a^{\frac{x}{n}}$$\frac{24 + 36\sqrt{10} + 10\sqrt{10} + 30\left(5^{\frac{1}{2}}\right)^{2}}{ - 43}$
Apply the power rule and multiply exponents, $\left(a^{m}\right)^{n} = a^{mn}$$\frac{24 + 36\sqrt{10} + 10\sqrt{10} + 30 \cdot 5^{\frac{1}{2} \cdot 2}}{ - 43}$
Combine $\frac{1}{2}$$\frac{24 + 36\sqrt{10} + 10\sqrt{10} + 30 \cdot 5^{\frac{2}{2}}}{ - 43}$
Cancel the common factor$\frac{24 + 36\sqrt{10} + 10\sqrt{10} + 30 \cdot 5^{\frac{2}{2}}}{ - 43}$
Rewrite the expression$\frac{24 + 36\sqrt{10} + 10\sqrt{10} + 30 \cdot 5^{1}}{ - 43}$
Evaluate the exponent$\frac{24 + 36\sqrt{10} + 10\sqrt{10} + 30 \cdot 5}{ - 43}$
Multiply $30$$\frac{24 + 36\sqrt{10} + 10\sqrt{10} + 150}{ - 43}$
Add $24$$\frac{174 + 36\sqrt{10} + 10\sqrt{10}}{ - 43}$
Add $36\sqrt{10}$$\frac{174 + 46\sqrt{10}}{ - 43}$
Move the negative in front of the fraction$ - \frac{174 + 46\sqrt{10}}{43}$
Form: $ - 7$$42941331