Solve frac{2sqrt{5}+sqrt{10}}{4sqrt{5}-sqrt{10}}

$$\frac { 2 \sqrt { 5 } + \sqrt { 10 } } { 4 \sqrt { 5 } - \sqrt { 10 } }$$

Evaluate

$$\frac{3\sqrt{2}+5}{7}\approx 1.320377241$$

Solution Steps

Rationalize the denominator of \frac{2\sqrt{5}+\sqrt{10}}{4\sqrt{5}-\sqrt{10}} by multiplying numerator and denominator by 4\sqrt{5}+\sqrt{10}.

$$\frac{\left(2\sqrt{5}+\sqrt{10}\right)\left(4\sqrt{5}+\sqrt{10}\right)}{\left(4\sqrt{5}-\sqrt{10}\right)\left(4\sqrt{5}+\sqrt{10}\right)}$$

Consider \left(4\sqrt{5}-\sqrt{10}\right)\left(4\sqrt{5}+\sqrt{10}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(2\sqrt{5}+\sqrt{10}\right)\left(4\sqrt{5}+\sqrt{10}\right)}{\left(4\sqrt{5}\right)^{2}-\left(\sqrt{10}\right)^{2}}$$

Expand \left(4\sqrt{5}\right)^{2}.

$$\frac{\left(2\sqrt{5}+\sqrt{10}\right)\left(4\sqrt{5}+\sqrt{10}\right)}{4^{2}\left(\sqrt{5}\right)^{2}-\left(\sqrt{10}\right)^{2}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{\left(2\sqrt{5}+\sqrt{10}\right)\left(4\sqrt{5}+\sqrt{10}\right)}{16\left(\sqrt{5}\right)^{2}-\left(\sqrt{10}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{\left(2\sqrt{5}+\sqrt{10}\right)\left(4\sqrt{5}+\sqrt{10}\right)}{16\times 5-\left(\sqrt{10}\right)^{2}}$$

Multiply 16 and 5 to get 80.

$$\frac{\left(2\sqrt{5}+\sqrt{10}\right)\left(4\sqrt{5}+\sqrt{10}\right)}{80-\left(\sqrt{10}\right)^{2}}$$

The square of \sqrt{10} is 10.

$$\frac{\left(2\sqrt{5}+\sqrt{10}\right)\left(4\sqrt{5}+\sqrt{10}\right)}{80-10}$$

Subtract 10 from 80 to get 70.

$$\frac{\left(2\sqrt{5}+\sqrt{10}\right)\left(4\sqrt{5}+\sqrt{10}\right)}{70}$$

Apply the distributive property by multiplying each term of 2\sqrt{5}+\sqrt{10} by each term of 4\sqrt{5}+\sqrt{10}.

$$\frac{8\left(\sqrt{5}\right)^{2}+2\sqrt{5}\sqrt{10}+4\sqrt{10}\sqrt{5}+\left(\sqrt{10}\right)^{2}}{70}$$

The square of \sqrt{5} is 5.

$$\frac{8\times 5+2\sqrt{5}\sqrt{10}+4\sqrt{10}\sqrt{5}+\left(\sqrt{10}\right)^{2}}{70}$$

Multiply 8 and 5 to get 40.

$$\frac{40+2\sqrt{5}\sqrt{10}+4\sqrt{10}\sqrt{5}+\left(\sqrt{10}\right)^{2}}{70}$$

Factor 10=5\times 2. Rewrite the square root of the product \sqrt{5\times 2} as the product of square roots \sqrt{5}\sqrt{2}.

$$\frac{40+2\sqrt{5}\sqrt{5}\sqrt{2}+4\sqrt{10}\sqrt{5}+\left(\sqrt{10}\right)^{2}}{70}$$

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

$$\frac{40+2\times 5\sqrt{2}+4\sqrt{10}\sqrt{5}+\left(\sqrt{10}\right)^{2}}{70}$$

Multiply 2 and 5 to get 10.

$$\frac{40+10\sqrt{2}+4\sqrt{10}\sqrt{5}+\left(\sqrt{10}\right)^{2}}{70}$$

Factor 10=5\times 2. Rewrite the square root of the product \sqrt{5\times 2} as the product of square roots \sqrt{5}\sqrt{2}.

$$\frac{40+10\sqrt{2}+4\sqrt{5}\sqrt{2}\sqrt{5}+\left(\sqrt{10}\right)^{2}}{70}$$

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

$$\frac{40+10\sqrt{2}+4\times 5\sqrt{2}+\left(\sqrt{10}\right)^{2}}{70}$$

Multiply 4 and 5 to get 20.

$$\frac{40+10\sqrt{2}+20\sqrt{2}+\left(\sqrt{10}\right)^{2}}{70}$$

Combine 10\sqrt{2} and 20\sqrt{2} to get 30\sqrt{2}.

$$\frac{40+30\sqrt{2}+\left(\sqrt{10}\right)^{2}}{70}$$

The square of \sqrt{10} is 10.

$$\frac{40+30\sqrt{2}+10}{70}$$

Add 40 and 10 to get 50.

$$\frac{50+30\sqrt{2}}{70}$$