Solve frac{2sqrt{40}-8sqrt{2}}{2sqrt{10}+2sqrt{18}}

$$\frac{ 2 \sqrt{ 40 } -8 \sqrt{ 2 } }{ 2 \sqrt{ 10 } +2 \sqrt{ 18 } }$$

Evaluate

$$\frac{5\sqrt{5}-11}{2}\approx 0.090169944$$

Solution Steps

Factor 40=2^{2}\times 10. Rewrite the square root of the product \sqrt{2^{2}\times 10} as the product of square roots \sqrt{2^{2}}\sqrt{10}. Take the square root of 2^{2}.

$$\frac{2\times 2\sqrt{10}-8\sqrt{2}}{2\sqrt{10}+2\sqrt{18}}$$

Multiply 2 and 2 to get 4.

$$\frac{4\sqrt{10}-8\sqrt{2}}{2\sqrt{10}+2\sqrt{18}}$$

Factor 18=3^{2}\times 2. Rewrite the square root of the product \sqrt{3^{2}\times 2} as the product of square roots \sqrt{3^{2}}\sqrt{2}. Take the square root of 3^{2}.

$$\frac{4\sqrt{10}-8\sqrt{2}}{2\sqrt{10}+2\times 3\sqrt{2}}$$

Multiply 2 and 3 to get 6.

$$\frac{4\sqrt{10}-8\sqrt{2}}{2\sqrt{10}+6\sqrt{2}}$$

Rationalize the denominator of \frac{4\sqrt{10}-8\sqrt{2}}{2\sqrt{10}+6\sqrt{2}} by multiplying numerator and denominator by 2\sqrt{10}-6\sqrt{2}.

$$\frac{\left(4\sqrt{10}-8\sqrt{2}\right)\left(2\sqrt{10}-6\sqrt{2}\right)}{\left(2\sqrt{10}+6\sqrt{2}\right)\left(2\sqrt{10}-6\sqrt{2}\right)}$$

Consider \left(2\sqrt{10}+6\sqrt{2}\right)\left(2\sqrt{10}-6\sqrt{2}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(4\sqrt{10}-8\sqrt{2}\right)\left(2\sqrt{10}-6\sqrt{2}\right)}{\left(2\sqrt{10}\right)^{2}-\left(6\sqrt{2}\right)^{2}}$$

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

$$\frac{\left(4\sqrt{10}-8\sqrt{2}\right)\left(2\sqrt{10}-6\sqrt{2}\right)}{2^{2}\left(\sqrt{10}\right)^{2}-\left(6\sqrt{2}\right)^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{\left(4\sqrt{10}-8\sqrt{2}\right)\left(2\sqrt{10}-6\sqrt{2}\right)}{4\left(\sqrt{10}\right)^{2}-\left(6\sqrt{2}\right)^{2}}$$

The square of \sqrt{10} is 10.

$$\frac{\left(4\sqrt{10}-8\sqrt{2}\right)\left(2\sqrt{10}-6\sqrt{2}\right)}{4\times 10-\left(6\sqrt{2}\right)^{2}}$$

Multiply 4 and 10 to get 40.

$$\frac{\left(4\sqrt{10}-8\sqrt{2}\right)\left(2\sqrt{10}-6\sqrt{2}\right)}{40-\left(6\sqrt{2}\right)^{2}}$$

Expand \left(6\sqrt{2}\right)^{2}.

$$\frac{\left(4\sqrt{10}-8\sqrt{2}\right)\left(2\sqrt{10}-6\sqrt{2}\right)}{40-6^{2}\left(\sqrt{2}\right)^{2}}$$

Calculate 6 to the power of 2 and get 36.

$$\frac{\left(4\sqrt{10}-8\sqrt{2}\right)\left(2\sqrt{10}-6\sqrt{2}\right)}{40-36\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\left(4\sqrt{10}-8\sqrt{2}\right)\left(2\sqrt{10}-6\sqrt{2}\right)}{40-36\times 2}$$

Multiply 36 and 2 to get 72.

$$\frac{\left(4\sqrt{10}-8\sqrt{2}\right)\left(2\sqrt{10}-6\sqrt{2}\right)}{40-72}$$

Subtract 72 from 40 to get -32.

$$\frac{\left(4\sqrt{10}-8\sqrt{2}\right)\left(2\sqrt{10}-6\sqrt{2}\right)}{-32}$$

Apply the distributive property by multiplying each term of 4\sqrt{10}-8\sqrt{2} by each term of 2\sqrt{10}-6\sqrt{2}.

$$\frac{8\left(\sqrt{10}\right)^{2}-24\sqrt{2}\sqrt{10}-16\sqrt{10}\sqrt{2}+48\left(\sqrt{2}\right)^{2}}{-32}$$

The square of \sqrt{10} is 10.

$$\frac{8\times 10-24\sqrt{2}\sqrt{10}-16\sqrt{10}\sqrt{2}+48\left(\sqrt{2}\right)^{2}}{-32}$$

Multiply 8 and 10 to get 80.

$$\frac{80-24\sqrt{2}\sqrt{10}-16\sqrt{10}\sqrt{2}+48\left(\sqrt{2}\right)^{2}}{-32}$$

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

$$\frac{80-24\sqrt{2}\sqrt{2}\sqrt{5}-16\sqrt{10}\sqrt{2}+48\left(\sqrt{2}\right)^{2}}{-32}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$\frac{80-24\times 2\sqrt{5}-16\sqrt{10}\sqrt{2}+48\left(\sqrt{2}\right)^{2}}{-32}$$

Multiply -24 and 2 to get -48.

$$\frac{80-48\sqrt{5}-16\sqrt{10}\sqrt{2}+48\left(\sqrt{2}\right)^{2}}{-32}$$

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

$$\frac{80-48\sqrt{5}-16\sqrt{2}\sqrt{5}\sqrt{2}+48\left(\sqrt{2}\right)^{2}}{-32}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$\frac{80-48\sqrt{5}-16\times 2\sqrt{5}+48\left(\sqrt{2}\right)^{2}}{-32}$$

Multiply -16 and 2 to get -32.

$$\frac{80-48\sqrt{5}-32\sqrt{5}+48\left(\sqrt{2}\right)^{2}}{-32}$$

Combine -48\sqrt{5} and -32\sqrt{5} to get -80\sqrt{5}.

$$\frac{80-80\sqrt{5}+48\left(\sqrt{2}\right)^{2}}{-32}$$

The square of \sqrt{2} is 2.

$$\frac{80-80\sqrt{5}+48\times 2}{-32}$$

Multiply 48 and 2 to get 96.

$$\frac{80-80\sqrt{5}+96}{-32}$$

Add 80 and 96 to get 176.

$$\frac{176-80\sqrt{5}}{-32}$$

Factor

$$\frac{5 \sqrt{5} - 11}{2} = 0.09016994374947451$$