Solve (sqrt{27}-sqrt{1/5})-(sqrt{45}+sqrt{48})

$$( \sqrt { 27 } - \sqrt { \frac { 1 } { 5 } } ) - ( \sqrt { 45 } + \sqrt { 48 } )$$

Evaluate

$$-\frac{16\sqrt{5}}{5}-\sqrt{3}\approx -8.887468336$$

Short Solution Steps

Factor 27=3^{2}\times 3. Rewrite the square root of the product \sqrt{3^{2}\times 3} as the product of square roots \sqrt{3^{2}}\sqrt{3}. Take the square root of 3^{2}.

$$3\sqrt{3}-\sqrt{\frac{1}{5}}-\left(\sqrt{45}+\sqrt{48}\right)$$

Rewrite the square root of the division \sqrt{\frac{1}{5}} as the division of square roots \frac{\sqrt{1}}{\sqrt{5}}.

$$3\sqrt{3}-\frac{\sqrt{1}}{\sqrt{5}}-\left(\sqrt{45}+\sqrt{48}\right)$$

Calculate the square root of 1 and get 1.

$$3\sqrt{3}-\frac{1}{\sqrt{5}}-\left(\sqrt{45}+\sqrt{48}\right)$$

Rationalize the denominator of \frac{1}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$3\sqrt{3}-\frac{\sqrt{5}}{\left(\sqrt{5}\right)^{2}}-\left(\sqrt{45}+\sqrt{48}\right)$$

The square of \sqrt{5} is 5.

$$3\sqrt{3}-\frac{\sqrt{5}}{5}-\left(\sqrt{45}+\sqrt{48}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 3\sqrt{3} times \frac{5}{5}.

$$\frac{5\times 3\sqrt{3}}{5}-\frac{\sqrt{5}}{5}-\left(\sqrt{45}+\sqrt{48}\right)$$

Since \frac{5\times 3\sqrt{3}}{5} and \frac{\sqrt{5}}{5} have the same denominator, subtract them by subtracting their numerators.

$$\frac{5\times 3\sqrt{3}-\sqrt{5}}{5}-\left(\sqrt{45}+\sqrt{48}\right)$$

Do the multiplications in 5\times 3\sqrt{3}-\sqrt{5}.

$$\frac{15\sqrt{3}-\sqrt{5}}{5}-\left(\sqrt{45}+\sqrt{48}\right)$$

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}.

$$\frac{15\sqrt{3}-\sqrt{5}}{5}-\left(3\sqrt{5}+\sqrt{48}\right)$$

Factor 48=4^{2}\times 3. Rewrite the square root of the product \sqrt{4^{2}\times 3} as the product of square roots \sqrt{4^{2}}\sqrt{3}. Take the square root of 4^{2}.

$$\frac{15\sqrt{3}-\sqrt{5}}{5}-\left(3\sqrt{5}+4\sqrt{3}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 3\sqrt{5}+4\sqrt{3} times \frac{5}{5}.

$$\frac{15\sqrt{3}-\sqrt{5}}{5}-\frac{5\left(3\sqrt{5}+4\sqrt{3}\right)}{5}$$

Since \frac{15\sqrt{3}-\sqrt{5}}{5} and \frac{5\left(3\sqrt{5}+4\sqrt{3}\right)}{5} have the same denominator, subtract them by subtracting their numerators.

$$\frac{15\sqrt{3}-\sqrt{5}-5\left(3\sqrt{5}+4\sqrt{3}\right)}{5}$$

Do the multiplications in 15\sqrt{3}-\sqrt{5}-5\left(3\sqrt{5}+4\sqrt{3}\right).

$$\frac{15\sqrt{3}-\sqrt{5}-15\sqrt{5}-20\sqrt{3}}{5}$$

Do the calculations in 15\sqrt{3}-\sqrt{5}-15\sqrt{5}-20\sqrt{3}.

$$\frac{-5\sqrt{3}-16\sqrt{5}}{5}$$