Solve (3)(sqrt{45}+sqrt{27})-(sqrt{4/3}+sqrt{125})

$$( 3 ) ( \sqrt { 45 } + \sqrt { 27 } ) - ( \sqrt { \frac { 4 } { 3 } } + \sqrt { 125 } )$$

Evaluate

$$\frac{25\sqrt{3}}{3}+4\sqrt{5}\approx 23.37802864$$

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}.

$$3\left(3\sqrt{5}+\sqrt{27}\right)-\left(\sqrt{\frac{4}{3}}+\sqrt{125}\right)$$

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}.

$$3\left(3\sqrt{5}+3\sqrt{3}\right)-\left(\sqrt{\frac{4}{3}}+\sqrt{125}\right)$$

Use the distributive property to multiply 3 by 3\sqrt{5}+3\sqrt{3}.

$$9\sqrt{5}+9\sqrt{3}-\left(\sqrt{\frac{4}{3}}+\sqrt{125}\right)$$

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

$$9\sqrt{5}+9\sqrt{3}-\left(\frac{\sqrt{4}}{\sqrt{3}}+\sqrt{125}\right)$$

Calculate the square root of 4 and get 2.

$$9\sqrt{5}+9\sqrt{3}-\left(\frac{2}{\sqrt{3}}+\sqrt{125}\right)$$

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

$$9\sqrt{5}+9\sqrt{3}-\left(\frac{2\sqrt{3}}{\left(\sqrt{3}\right)^{2}}+\sqrt{125}\right)$$

The square of \sqrt{3} is 3.

$$9\sqrt{5}+9\sqrt{3}-\left(\frac{2\sqrt{3}}{3}+\sqrt{125}\right)$$

Factor 125=5^{2}\times 5. Rewrite the square root of the product \sqrt{5^{2}\times 5} as the product of square roots \sqrt{5^{2}}\sqrt{5}. Take the square root of 5^{2}.

$$9\sqrt{5}+9\sqrt{3}-\left(\frac{2\sqrt{3}}{3}+5\sqrt{5}\right)$$

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

$$9\sqrt{5}+9\sqrt{3}-\left(\frac{2\sqrt{3}}{3}+\frac{3\times 5\sqrt{5}}{3}\right)$$

Since \frac{2\sqrt{3}}{3} and \frac{3\times 5\sqrt{5}}{3} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 2\sqrt{3}+3\times 5\sqrt{5}.

$$9\sqrt{5}+9\sqrt{3}-\frac{2\sqrt{3}+15\sqrt{5}}{3}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 9\sqrt{5}+9\sqrt{3} times \frac{3}{3}.

$$\frac{3\left(9\sqrt{5}+9\sqrt{3}\right)}{3}-\frac{2\sqrt{3}+15\sqrt{5}}{3}$$

Since \frac{3\left(9\sqrt{5}+9\sqrt{3}\right)}{3} and \frac{2\sqrt{3}+15\sqrt{5}}{3} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3\left(9\sqrt{5}+9\sqrt{3}\right)-\left(2\sqrt{3}+15\sqrt{5}\right)}{3}$$

Do the multiplications in 3\left(9\sqrt{5}+9\sqrt{3}\right)-\left(2\sqrt{3}+15\sqrt{5}\right).

$$\frac{27\sqrt{5}+27\sqrt{3}-2\sqrt{3}-15\sqrt{5}}{3}$$

Do the calculations in 27\sqrt{5}+27\sqrt{3}-2\sqrt{3}-15\sqrt{5}.

$$\frac{12\sqrt{5}+25\sqrt{3}}{3}$$