Solve 2-3/sqrt{5}-2/sqrt{45}+6/sqrt{80}

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Arithmetic

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2-\frac{3\sqrt{5}}{\left(\sqrt{5}\right)^{2}}-\frac{2}{\sqrt{45}}+\frac{6}{\sqrt{80}}

Rationalize the denominator of \frac{3}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

2-\frac{3\sqrt{5}}{5}-\frac{2}{\sqrt{45}}+\frac{6}{\sqrt{80}}

The square of \sqrt{5} is 5.

2-\frac{3\sqrt{5}}{5}-\frac{2}{3\sqrt{5}}+\frac{6}{\sqrt{80}}

Factor 45=3^{2}\times 5. Rewrite the square root of the product \sqrt{3^{2}\times 5} as the product of square roots \sqrt{3^{2}}\sqrt{5}. Take the square root of 3^{2}.

2-\frac{3\sqrt{5}}{5}-\frac{2\sqrt{5}}{3\left(\sqrt{5}\right)^{2}}+\frac{6}{\sqrt{80}}

Rationalize the denominator of \frac{2}{3\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

2-\frac{3\sqrt{5}}{5}-\frac{2\sqrt{5}}{3\times 5}+\frac{6}{\sqrt{80}}

The square of \sqrt{5} is 5.

2-\frac{3\sqrt{5}}{5}-\frac{2\sqrt{5}}{15}+\frac{6}{\sqrt{80}}

Multiply 3 and 5 to get 15.

2-\frac{11}{15}\sqrt{5}+\frac{6}{\sqrt{80}}

Combine -\frac{3\sqrt{5}}{5} and -\frac{2\sqrt{5}}{15} to get -\frac{11}{15}\sqrt{5}.

2-\frac{11}{15}\sqrt{5}+\frac{6}{4\sqrt{5}}

Factor 80=4^{2}\times 5. Rewrite the square root of the product \sqrt{4^{2}\times 5} as the product of square roots \sqrt{4^{2}}\sqrt{5}. Take the square root of 4^{2}.

2-\frac{11}{15}\sqrt{5}+\frac{6\sqrt{5}}{4\left(\sqrt{5}\right)^{2}}

Rationalize the denominator of \frac{6}{4\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

2-\frac{11}{15}\sqrt{5}+\frac{6\sqrt{5}}{4\times 5}

The square of \sqrt{5} is 5.

2-\frac{11}{15}\sqrt{5}+\frac{3\sqrt{5}}{2\times 5}

Cancel out 2 in both numerator and denominator.

2-\frac{11}{15}\sqrt{5}+\frac{3\sqrt{5}}{10}

Multiply 2 and 5 to get 10.

2-\frac{13}{30}\sqrt{5}

Combine -\frac{11}{15}\sqrt{5} and \frac{3\sqrt{5}}{10} to get -\frac{13}{30}\sqrt{5}.