Solve (sqrt{3^2*5^2}-sqrt{50}:sqrt{2}):5+(15+sqrt{7^2*3^2}):(14+sqrt{15})

$$( \sqrt { 3 ^ { 2 } \times 5 ^ { 2 } } - \sqrt { 50 } : \sqrt { 2 } ) : 5 + ( 15 + \sqrt { 7 ^ { 2 } \times 3 ^ { 2 } } ) : ( 14 + \sqrt { 15 } )$$

Evaluate

$$\frac{866-36\sqrt{15}}{181}\approx 4.014213257$$

Short Solution Steps

Calculate 3 to the power of 2 and get 9.

$$\frac{\sqrt{9\times 5^{2}}-\frac{\sqrt{50}}{\sqrt{2}}}{5}+\frac{15+\sqrt{7^{2}\times 3^{2}}}{14+\sqrt{15}}$$

Calculate 5 to the power of 2 and get 25.

$$\frac{\sqrt{9\times 25}-\frac{\sqrt{50}}{\sqrt{2}}}{5}+\frac{15+\sqrt{7^{2}\times 3^{2}}}{14+\sqrt{15}}$$

Multiply 9 and 25 to get 225.

$$\frac{\sqrt{225}-\frac{\sqrt{50}}{\sqrt{2}}}{5}+\frac{15+\sqrt{7^{2}\times 3^{2}}}{14+\sqrt{15}}$$

Calculate the square root of 225 and get 15.

$$\frac{15-\frac{\sqrt{50}}{\sqrt{2}}}{5}+\frac{15+\sqrt{7^{2}\times 3^{2}}}{14+\sqrt{15}}$$

Rewrite the division of square roots \frac{\sqrt{50}}{\sqrt{2}} as the square root of the division \sqrt{\frac{50}{2}} and perform the division.

$$\frac{15-\sqrt{25}}{5}+\frac{15+\sqrt{7^{2}\times 3^{2}}}{14+\sqrt{15}}$$

Calculate the square root of 25 and get 5.

$$\frac{15-5}{5}+\frac{15+\sqrt{7^{2}\times 3^{2}}}{14+\sqrt{15}}$$

Subtract 5 from 15 to get 10.

$$\frac{10}{5}+\frac{15+\sqrt{7^{2}\times 3^{2}}}{14+\sqrt{15}}$$

Divide 10 by 5 to get 2.

$$2+\frac{15+\sqrt{7^{2}\times 3^{2}}}{14+\sqrt{15}}$$

Calculate 7 to the power of 2 and get 49.

$$2+\frac{15+\sqrt{49\times 3^{2}}}{14+\sqrt{15}}$$

Calculate 3 to the power of 2 and get 9.

$$2+\frac{15+\sqrt{49\times 9}}{14+\sqrt{15}}$$

Multiply 49 and 9 to get 441.

$$2+\frac{15+\sqrt{441}}{14+\sqrt{15}}$$

Calculate the square root of 441 and get 21.

$$2+\frac{15+21}{14+\sqrt{15}}$$

Add 15 and 21 to get 36.

$$2+\frac{36}{14+\sqrt{15}}$$

Rationalize the denominator of \frac{36}{14+\sqrt{15}} by multiplying numerator and denominator by 14-\sqrt{15}.

$$2+\frac{36\left(14-\sqrt{15}\right)}{\left(14+\sqrt{15}\right)\left(14-\sqrt{15}\right)}$$

Consider \left(14+\sqrt{15}\right)\left(14-\sqrt{15}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$2+\frac{36\left(14-\sqrt{15}\right)}{14^{2}-\left(\sqrt{15}\right)^{2}}$$

Square 14. Square \sqrt{15}.

$$2+\frac{36\left(14-\sqrt{15}\right)}{196-15}$$

Subtract 15 from 196 to get 181.

$$2+\frac{36\left(14-\sqrt{15}\right)}{181}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 2 times \frac{181}{181}.

$$\frac{2\times 181}{181}+\frac{36\left(14-\sqrt{15}\right)}{181}$$

Since \frac{2\times 181}{181} and \frac{36\left(14-\sqrt{15}\right)}{181} have the same denominator, add them by adding their numerators.

$$\frac{2\times 181+36\left(14-\sqrt{15}\right)}{181}$$

Do the multiplications in 2\times 181+36\left(14-\sqrt{15}\right).

$$\frac{362+504-36\sqrt{15}}{181}$$

Do the calculations in 362+504-36\sqrt{15}.

$$\frac{866-36\sqrt{15}}{181}$$

Factor

$$\frac{2 {(433 - 18 \sqrt{15})}}{181} = 4.014213257107917$$