Solve (sqrt{45}+sqrt{27})-(sqrt{4/3}+sqrt{35})

$$( \sqrt { 45 } + \sqrt { 27 } ) - ( \sqrt { \frac { 4 } { 3 } } + \sqrt { 35 } )$$

Evaluate

$$\frac{7\sqrt{3}}{3}+3\sqrt{5}-\sqrt{35}\approx 4.833576034$$

Short Solution Steps

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}.

$$3\sqrt{5}+\sqrt{27}-\left(\sqrt{\frac{4}{3}}+\sqrt{35}\right)$$

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}.

$$3\sqrt{5}+3\sqrt{3}-\left(\sqrt{\frac{4}{3}}+\sqrt{35}\right)$$

Rewrite the square root of the division \sqrt{\frac{4}{3}} as the division of square roots \frac{\sqrt{4}}{\sqrt{3}}.

$$3\sqrt{5}+3\sqrt{3}-\left(\frac{\sqrt{4}}{\sqrt{3}}+\sqrt{35}\right)$$

Calculate the square root of 4 and get 2.

$$3\sqrt{5}+3\sqrt{3}-\left(\frac{2}{\sqrt{3}}+\sqrt{35}\right)$$

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

$$3\sqrt{5}+3\sqrt{3}-\left(\frac{2\sqrt{3}}{\left(\sqrt{3}\right)^{2}}+\sqrt{35}\right)$$

The square of \sqrt{3} is 3.

$$3\sqrt{5}+3\sqrt{3}-\left(\frac{2\sqrt{3}}{3}+\sqrt{35}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply \sqrt{35} times \frac{3}{3}.

$$3\sqrt{5}+3\sqrt{3}-\left(\frac{2\sqrt{3}}{3}+\frac{3\sqrt{35}}{3}\right)$$

Since \frac{2\sqrt{3}}{3} and \frac{3\sqrt{35}}{3} have the same denominator, add them by adding their numerators.

$$3\sqrt{5}+3\sqrt{3}-\frac{2\sqrt{3}+3\sqrt{35}}{3}$$

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

$$\frac{3\left(3\sqrt{5}+3\sqrt{3}\right)}{3}-\frac{2\sqrt{3}+3\sqrt{35}}{3}$$

Since \frac{3\left(3\sqrt{5}+3\sqrt{3}\right)}{3} and \frac{2\sqrt{3}+3\sqrt{35}}{3} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3\left(3\sqrt{5}+3\sqrt{3}\right)-\left(2\sqrt{3}+3\sqrt{35}\right)}{3}$$

Do the multiplications in 3\left(3\sqrt{5}+3\sqrt{3}\right)-\left(2\sqrt{3}+3\sqrt{35}\right).

$$\frac{9\sqrt{5}+9\sqrt{3}-2\sqrt{3}-3\sqrt{35}}{3}$$

Do the calculations in 9\sqrt{5}+9\sqrt{3}-2\sqrt{3}-3\sqrt{35}.

$$\frac{9\sqrt{5}-3\sqrt{35}+7\sqrt{3}}{3}$$