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

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

Evaluate

$$\frac{23\sqrt{5}}{30}\approx 1.714318783$$

Solution Steps

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

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

The square of \sqrt{5} is 5.

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

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}.

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

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

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

The square of \sqrt{5} is 5.

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

Multiply 3 and 5 to get 15.

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

Combine \frac{3\sqrt{5}}{5} and -\frac{2\sqrt{5}}{15} to get \frac{7}{15}\sqrt{5}.

$$\frac{7}{15}\sqrt{5}+\frac{6}{\sqrt{80}}$$

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}.

$$\frac{7}{15}\sqrt{5}+\frac{6}{4\sqrt{5}}$$

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

$$\frac{7}{15}\sqrt{5}+\frac{6\sqrt{5}}{4\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{7}{15}\sqrt{5}+\frac{6\sqrt{5}}{4\times 5}$$

Cancel out 2 in both numerator and denominator.

$$\frac{7}{15}\sqrt{5}+\frac{3\sqrt{5}}{2\times 5}$$

Multiply 2 and 5 to get 10.

$$\frac{7}{15}\sqrt{5}+\frac{3\sqrt{5}}{10}$$

Combine \frac{7}{15}\sqrt{5} and \frac{3\sqrt{5}}{10} to get \frac{23}{30}\sqrt{5}.

$$\frac{23}{30}\sqrt{5}$$