# Evaluate: (3sqrt32+2sqrt125)/(sqrt2-3sqrt5)

## Expression: (3sqrt32+2sqrt125)/(sqrt2-3sqrt5)

Write the problem as a mathematical expression

$\frac{3\sqrt{32} + 2\sqrt{125}}{\sqrt{2} - 3\sqrt{5}}$

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: $- 7$

$42941331$

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