Solve frac{sqrt{25}+sqrt{45}}{sqrt{45}-sqrt{30}}

$$\frac { \sqrt { 25 } + \sqrt { 45 } } { \sqrt { 45 } - \sqrt { 30 } }$$

Evaluate

$$\frac{\sqrt{30}}{3}+\sqrt{5}+\sqrt{6}+3\approx 9.511299579$$

Solution Steps

Calculate the square root of 25 and get 5.

$$\frac{5+\sqrt{45}}{\sqrt{45}-\sqrt{30}}$$

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}.

$$\frac{5+3\sqrt{5}}{\sqrt{45}-\sqrt{30}}$$

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}.

$$\frac{5+3\sqrt{5}}{3\sqrt{5}-\sqrt{30}}$$

Rationalize the denominator of \frac{5+3\sqrt{5}}{3\sqrt{5}-\sqrt{30}} by multiplying numerator and denominator by 3\sqrt{5}+\sqrt{30}.

$$\frac{\left(5+3\sqrt{5}\right)\left(3\sqrt{5}+\sqrt{30}\right)}{\left(3\sqrt{5}-\sqrt{30}\right)\left(3\sqrt{5}+\sqrt{30}\right)}$$

Consider \left(3\sqrt{5}-\sqrt{30}\right)\left(3\sqrt{5}+\sqrt{30}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(5+3\sqrt{5}\right)\left(3\sqrt{5}+\sqrt{30}\right)}{\left(3\sqrt{5}\right)^{2}-\left(\sqrt{30}\right)^{2}}$$

Expand \left(3\sqrt{5}\right)^{2}.

$$\frac{\left(5+3\sqrt{5}\right)\left(3\sqrt{5}+\sqrt{30}\right)}{3^{2}\left(\sqrt{5}\right)^{2}-\left(\sqrt{30}\right)^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{\left(5+3\sqrt{5}\right)\left(3\sqrt{5}+\sqrt{30}\right)}{9\left(\sqrt{5}\right)^{2}-\left(\sqrt{30}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{\left(5+3\sqrt{5}\right)\left(3\sqrt{5}+\sqrt{30}\right)}{9\times 5-\left(\sqrt{30}\right)^{2}}$$

Multiply 9 and 5 to get 45.

$$\frac{\left(5+3\sqrt{5}\right)\left(3\sqrt{5}+\sqrt{30}\right)}{45-\left(\sqrt{30}\right)^{2}}$$

The square of \sqrt{30} is 30.

$$\frac{\left(5+3\sqrt{5}\right)\left(3\sqrt{5}+\sqrt{30}\right)}{45-30}$$

Subtract 30 from 45 to get 15.

$$\frac{\left(5+3\sqrt{5}\right)\left(3\sqrt{5}+\sqrt{30}\right)}{15}$$

Apply the distributive property by multiplying each term of 5+3\sqrt{5} by each term of 3\sqrt{5}+\sqrt{30}.

$$\frac{15\sqrt{5}+5\sqrt{30}+9\left(\sqrt{5}\right)^{2}+3\sqrt{5}\sqrt{30}}{15}$$

The square of \sqrt{5} is 5.

$$\frac{15\sqrt{5}+5\sqrt{30}+9\times 5+3\sqrt{5}\sqrt{30}}{15}$$

Multiply 9 and 5 to get 45.

$$\frac{15\sqrt{5}+5\sqrt{30}+45+3\sqrt{5}\sqrt{30}}{15}$$

Factor 30=5\times 6. Rewrite the square root of the product \sqrt{5\times 6} as the product of square roots \sqrt{5}\sqrt{6}.

$$\frac{15\sqrt{5}+5\sqrt{30}+45+3\sqrt{5}\sqrt{5}\sqrt{6}}{15}$$

Multiply \sqrt{5} and \sqrt{5} to get 5.

$$\frac{15\sqrt{5}+5\sqrt{30}+45+3\times 5\sqrt{6}}{15}$$

Multiply 3 and 5 to get 15.

$$\frac{15\sqrt{5}+5\sqrt{30}+45+15\sqrt{6}}{15}$$