Solve frac{2sqrt{40}-8sqrt{2}}{2sqrt{10}+2sqrt{182}}

$$\frac { 2 \sqrt { 40 } - 8 \sqrt { 2 } } { 2 \sqrt { 10 } + 2 \sqrt { 182 } }$$

Evaluate

$$\frac{\sqrt{455}+2\sqrt{5}-2\sqrt{91}-5}{43}\approx 0.040094905$$

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}.

$$\frac{2\times 2\sqrt{10}-8\sqrt{2}}{2\sqrt{10}+2\sqrt{182}}$$

Multiply 2 and 2 to get 4.

$$\frac{4\sqrt{10}-8\sqrt{2}}{2\sqrt{10}+2\sqrt{182}}$$

Rationalize the denominator of \frac{4\sqrt{10}-8\sqrt{2}}{2\sqrt{10}+2\sqrt{182}} by multiplying numerator and denominator by 2\sqrt{10}-2\sqrt{182}.

$$\frac{\left(4\sqrt{10}-8\sqrt{2}\right)\left(2\sqrt{10}-2\sqrt{182}\right)}{\left(2\sqrt{10}+2\sqrt{182}\right)\left(2\sqrt{10}-2\sqrt{182}\right)}$$

Consider \left(2\sqrt{10}+2\sqrt{182}\right)\left(2\sqrt{10}-2\sqrt{182}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(4\sqrt{10}-8\sqrt{2}\right)\left(2\sqrt{10}-2\sqrt{182}\right)}{\left(2\sqrt{10}\right)^{2}-\left(2\sqrt{182}\right)^{2}}$$

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

$$\frac{\left(4\sqrt{10}-8\sqrt{2}\right)\left(2\sqrt{10}-2\sqrt{182}\right)}{2^{2}\left(\sqrt{10}\right)^{2}-\left(2\sqrt{182}\right)^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{\left(4\sqrt{10}-8\sqrt{2}\right)\left(2\sqrt{10}-2\sqrt{182}\right)}{4\left(\sqrt{10}\right)^{2}-\left(2\sqrt{182}\right)^{2}}$$

The square of \sqrt{10} is 10.

$$\frac{\left(4\sqrt{10}-8\sqrt{2}\right)\left(2\sqrt{10}-2\sqrt{182}\right)}{4\times 10-\left(2\sqrt{182}\right)^{2}}$$

Multiply 4 and 10 to get 40.

$$\frac{\left(4\sqrt{10}-8\sqrt{2}\right)\left(2\sqrt{10}-2\sqrt{182}\right)}{40-\left(2\sqrt{182}\right)^{2}}$$

Expand \left(2\sqrt{182}\right)^{2}.

$$\frac{\left(4\sqrt{10}-8\sqrt{2}\right)\left(2\sqrt{10}-2\sqrt{182}\right)}{40-2^{2}\left(\sqrt{182}\right)^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{\left(4\sqrt{10}-8\sqrt{2}\right)\left(2\sqrt{10}-2\sqrt{182}\right)}{40-4\left(\sqrt{182}\right)^{2}}$$

The square of \sqrt{182} is 182.

$$\frac{\left(4\sqrt{10}-8\sqrt{2}\right)\left(2\sqrt{10}-2\sqrt{182}\right)}{40-4\times 182}$$

Multiply 4 and 182 to get 728.

$$\frac{\left(4\sqrt{10}-8\sqrt{2}\right)\left(2\sqrt{10}-2\sqrt{182}\right)}{40-728}$$

Subtract 728 from 40 to get -688.

$$\frac{\left(4\sqrt{10}-8\sqrt{2}\right)\left(2\sqrt{10}-2\sqrt{182}\right)}{-688}$$

Apply the distributive property by multiplying each term of 4\sqrt{10}-8\sqrt{2} by each term of 2\sqrt{10}-2\sqrt{182}.

$$\frac{8\left(\sqrt{10}\right)^{2}-8\sqrt{10}\sqrt{182}-16\sqrt{10}\sqrt{2}+16\sqrt{2}\sqrt{182}}{-688}$$

The square of \sqrt{10} is 10.

$$\frac{8\times 10-8\sqrt{10}\sqrt{182}-16\sqrt{10}\sqrt{2}+16\sqrt{2}\sqrt{182}}{-688}$$

Multiply 8 and 10 to get 80.

$$\frac{80-8\sqrt{10}\sqrt{182}-16\sqrt{10}\sqrt{2}+16\sqrt{2}\sqrt{182}}{-688}$$

To multiply \sqrt{10} and \sqrt{182}, multiply the numbers under the square root.

$$\frac{80-8\sqrt{1820}-16\sqrt{10}\sqrt{2}+16\sqrt{2}\sqrt{182}}{-688}$$

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}.

$$\frac{80-8\sqrt{1820}-16\sqrt{2}\sqrt{5}\sqrt{2}+16\sqrt{2}\sqrt{182}}{-688}$$

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

$$\frac{80-8\sqrt{1820}-16\times 2\sqrt{5}+16\sqrt{2}\sqrt{182}}{-688}$$

Multiply -16 and 2 to get -32.

$$\frac{80-8\sqrt{1820}-32\sqrt{5}+16\sqrt{2}\sqrt{182}}{-688}$$

Factor 182=2\times 91. Rewrite the square root of the product \sqrt{2\times 91} as the product of square roots \sqrt{2}\sqrt{91}.

$$\frac{80-8\sqrt{1820}-32\sqrt{5}+16\sqrt{2}\sqrt{2}\sqrt{91}}{-688}$$

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

$$\frac{80-8\sqrt{1820}-32\sqrt{5}+16\times 2\sqrt{91}}{-688}$$

Multiply 16 and 2 to get 32.

$$\frac{80-8\sqrt{1820}-32\sqrt{5}+32\sqrt{91}}{-688}$$

Factor 1820=2^{2}\times 455. Rewrite the square root of the product \sqrt{2^{2}\times 455} as the product of square roots \sqrt{2^{2}}\sqrt{455}. Take the square root of 2^{2}.

$$\frac{80-8\times 2\sqrt{455}-32\sqrt{5}+32\sqrt{91}}{-688}$$

Multiply -8 and 2 to get -16.

$$\frac{80-16\sqrt{455}-32\sqrt{5}+32\sqrt{91}}{-688}$$

Factor

$$\frac{\sqrt{455} + 2 \sqrt{5} - 2 \sqrt{91} - 5}{43} = 0.04009490545028394$$