Solve 2sqrt{700}-15sqrt{1/45}+4sqrt{3/16}-56sqrt{1/3}

$$2 \sqrt { 700 } - 15 \sqrt { \frac { 1 } { 45 } } + 4 \sqrt { \frac { 3 } { 16 } } - 56 \sqrt { \frac { 1 } { 3 } }$$

Evaluate

$$-\frac{53\sqrt{3}}{3}+20\sqrt{7}-\sqrt{5}\approx 20.079393977$$

Short Solution Steps

Factor 700=10^{2}\times 7. Rewrite the square root of the product \sqrt{10^{2}\times 7} as the product of square roots \sqrt{10^{2}}\sqrt{7}. Take the square root of 10^{2}.

$$2\times 10\sqrt{7}-15\sqrt{\frac{1}{45}}+4\sqrt{\frac{3}{16}}-56\sqrt{\frac{1}{3}}$$

Multiply 2 and 10 to get 20.

$$20\sqrt{7}-15\sqrt{\frac{1}{45}}+4\sqrt{\frac{3}{16}}-56\sqrt{\frac{1}{3}}$$

Rewrite the square root of the division \sqrt{\frac{1}{45}} as the division of square roots \frac{\sqrt{1}}{\sqrt{45}}.

$$20\sqrt{7}-15\times \frac{\sqrt{1}}{\sqrt{45}}+4\sqrt{\frac{3}{16}}-56\sqrt{\frac{1}{3}}$$

Calculate the square root of 1 and get 1.

$$20\sqrt{7}-15\times \frac{1}{\sqrt{45}}+4\sqrt{\frac{3}{16}}-56\sqrt{\frac{1}{3}}$$

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}.

$$20\sqrt{7}-15\times \frac{1}{3\sqrt{5}}+4\sqrt{\frac{3}{16}}-56\sqrt{\frac{1}{3}}$$

Rationalize the denominator of \frac{1}{3\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$20\sqrt{7}-15\times \frac{\sqrt{5}}{3\left(\sqrt{5}\right)^{2}}+4\sqrt{\frac{3}{16}}-56\sqrt{\frac{1}{3}}$$

The square of \sqrt{5} is 5.

$$20\sqrt{7}-15\times \frac{\sqrt{5}}{3\times 5}+4\sqrt{\frac{3}{16}}-56\sqrt{\frac{1}{3}}$$

Multiply 3 and 5 to get 15.

$$20\sqrt{7}-15\times \frac{\sqrt{5}}{15}+4\sqrt{\frac{3}{16}}-56\sqrt{\frac{1}{3}}$$

Cancel out 15 and 15.

$$20\sqrt{7}-\sqrt{5}+4\sqrt{\frac{3}{16}}-56\sqrt{\frac{1}{3}}$$

Rewrite the square root of the division \sqrt{\frac{3}{16}} as the division of square roots \frac{\sqrt{3}}{\sqrt{16}}.

$$20\sqrt{7}-\sqrt{5}+4\times \frac{\sqrt{3}}{\sqrt{16}}-56\sqrt{\frac{1}{3}}$$

Calculate the square root of 16 and get 4.

$$20\sqrt{7}-\sqrt{5}+4\times \frac{\sqrt{3}}{4}-56\sqrt{\frac{1}{3}}$$

Cancel out 4 and 4.

$$20\sqrt{7}-\sqrt{5}+\sqrt{3}-56\sqrt{\frac{1}{3}}$$

Rewrite the square root of the division \sqrt{\frac{1}{3}} as the division of square roots \frac{\sqrt{1}}{\sqrt{3}}.

$$20\sqrt{7}-\sqrt{5}+\sqrt{3}-56\times \frac{\sqrt{1}}{\sqrt{3}}$$

Calculate the square root of 1 and get 1.

$$20\sqrt{7}-\sqrt{5}+\sqrt{3}-56\times \frac{1}{\sqrt{3}}$$

Rationalize the denominator of \frac{1}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$20\sqrt{7}-\sqrt{5}+\sqrt{3}-56\times \frac{\sqrt{3}}{\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$20\sqrt{7}-\sqrt{5}+\sqrt{3}-56\times \frac{\sqrt{3}}{3}$$

Express -56\times \frac{\sqrt{3}}{3} as a single fraction.

$$20\sqrt{7}-\sqrt{5}+\sqrt{3}+\frac{-56\sqrt{3}}{3}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 20\sqrt{7}-\sqrt{5}+\sqrt{3} times \frac{3}{3}.

$$\frac{3\left(20\sqrt{7}-\sqrt{5}+\sqrt{3}\right)}{3}+\frac{-56\sqrt{3}}{3}$$

Since \frac{3\left(20\sqrt{7}-\sqrt{5}+\sqrt{3}\right)}{3} and \frac{-56\sqrt{3}}{3} have the same denominator, add them by adding their numerators.

$$\frac{3\left(20\sqrt{7}-\sqrt{5}+\sqrt{3}\right)-56\sqrt{3}}{3}$$

Do the multiplications in 3\left(20\sqrt{7}-\sqrt{5}+\sqrt{3}\right)-56\sqrt{3}.

$$\frac{60\sqrt{7}-3\sqrt{5}+3\sqrt{3}-56\sqrt{3}}{3}$$

Do the calculations in 60\sqrt{7}-3\sqrt{5}+3\sqrt{3}-56\sqrt{3}.

$$\frac{60\sqrt{7}-3\sqrt{5}-53\sqrt{3}}{3}$$