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

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

Evaluate

$$4\sqrt{5}+20-6\sqrt{2}-6\sqrt{10}\approx 1.485324575$$

Solution Steps

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

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

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

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

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

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

Calculate 3 to the power of 2 and get 9.

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

The square of \sqrt{2} is 2.

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

Multiply 9 and 2 to get 18.

$$\frac{\left(4+4\sqrt{5}\right)\left(3\sqrt{2}-2\sqrt{5}\right)}{18-\left(2\sqrt{5}\right)^{2}}$$

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

$$\frac{\left(4+4\sqrt{5}\right)\left(3\sqrt{2}-2\sqrt{5}\right)}{18-2^{2}\left(\sqrt{5}\right)^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{\left(4+4\sqrt{5}\right)\left(3\sqrt{2}-2\sqrt{5}\right)}{18-4\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{\left(4+4\sqrt{5}\right)\left(3\sqrt{2}-2\sqrt{5}\right)}{18-4\times 5}$$

Multiply 4 and 5 to get 20.

$$\frac{\left(4+4\sqrt{5}\right)\left(3\sqrt{2}-2\sqrt{5}\right)}{18-20}$$

Subtract 20 from 18 to get -2.

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

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

$$\frac{12\sqrt{2}-8\sqrt{5}+12\sqrt{5}\sqrt{2}-8\left(\sqrt{5}\right)^{2}}{-2}$$

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

$$\frac{12\sqrt{2}-8\sqrt{5}+12\sqrt{10}-8\left(\sqrt{5}\right)^{2}}{-2}$$

The square of \sqrt{5} is 5.

$$\frac{12\sqrt{2}-8\sqrt{5}+12\sqrt{10}-8\times 5}{-2}$$

Multiply -8 and 5 to get -40.

$$\frac{12\sqrt{2}-8\sqrt{5}+12\sqrt{10}-40}{-2}$$

Factor

$$2 \sqrt{2} {(\sqrt{10} + 5 \sqrt{2} - 3 \sqrt{5} - 3)} = 1.485324575$$