Solve frac{sqrt{5}+4}{sqrt{2}+3sqrt{10}}

$$\frac { \sqrt { 5 } + 4 } { \sqrt { 2 } + 3 \sqrt { 10 } }$$

Evaluate

$$\frac{\sqrt{2}+\sqrt{10}}{8}\approx 0.572061403$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{5}+4}{\sqrt{2}+3\sqrt{10}} by multiplying numerator and denominator by \sqrt{2}-3\sqrt{10}.

$$\frac{\left(\sqrt{5}+4\right)\left(\sqrt{2}-3\sqrt{10}\right)}{\left(\sqrt{2}+3\sqrt{10}\right)\left(\sqrt{2}-3\sqrt{10}\right)}$$

Consider \left(\sqrt{2}+3\sqrt{10}\right)\left(\sqrt{2}-3\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}.

$$\frac{\left(\sqrt{5}+4\right)\left(\sqrt{2}-3\sqrt{10}\right)}{\left(\sqrt{2}\right)^{2}-\left(3\sqrt{10}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\left(\sqrt{5}+4\right)\left(\sqrt{2}-3\sqrt{10}\right)}{2-\left(3\sqrt{10}\right)^{2}}$$

Expand \left(3\sqrt{10}\right)^{2}.

$$\frac{\left(\sqrt{5}+4\right)\left(\sqrt{2}-3\sqrt{10}\right)}{2-3^{2}\left(\sqrt{10}\right)^{2}}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{\left(\sqrt{5}+4\right)\left(\sqrt{2}-3\sqrt{10}\right)}{2-9\left(\sqrt{10}\right)^{2}}$$

The square of \sqrt{10} is 10.

$$\frac{\left(\sqrt{5}+4\right)\left(\sqrt{2}-3\sqrt{10}\right)}{2-9\times 10}$$

Multiply 9 and 10 to get 90.

$$\frac{\left(\sqrt{5}+4\right)\left(\sqrt{2}-3\sqrt{10}\right)}{2-90}$$

Subtract 90 from 2 to get -88.

$$\frac{\left(\sqrt{5}+4\right)\left(\sqrt{2}-3\sqrt{10}\right)}{-88}$$

Apply the distributive property by multiplying each term of \sqrt{5}+4 by each term of \sqrt{2}-3\sqrt{10}.

$$\frac{\sqrt{5}\sqrt{2}-3\sqrt{5}\sqrt{10}+4\sqrt{2}-12\sqrt{10}}{-88}$$

To multiply \sqrt{5} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{\sqrt{10}-3\sqrt{5}\sqrt{10}+4\sqrt{2}-12\sqrt{10}}{-88}$$

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}.

$$\frac{\sqrt{10}-3\sqrt{5}\sqrt{5}\sqrt{2}+4\sqrt{2}-12\sqrt{10}}{-88}$$

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

$$\frac{\sqrt{10}-3\times 5\sqrt{2}+4\sqrt{2}-12\sqrt{10}}{-88}$$

Multiply -3 and 5 to get -15.

$$\frac{\sqrt{10}-15\sqrt{2}+4\sqrt{2}-12\sqrt{10}}{-88}$$

Combine -15\sqrt{2} and 4\sqrt{2} to get -11\sqrt{2}.

$$\frac{\sqrt{10}-11\sqrt{2}-12\sqrt{10}}{-88}$$

Combine \sqrt{10} and -12\sqrt{10} to get -11\sqrt{10}.

$$\frac{-11\sqrt{10}-11\sqrt{2}}{-88}$$