Solve frac{44-sqrt{2017}}{44-sqrt{2016}}

$$\frac{ 44- \sqrt{ 2017 } }{ 44- \sqrt{ 2016 } }$$

Evaluate

$$\frac{3\sqrt{28238}}{20}+\frac{11\sqrt{2017}}{20}-\frac{33\sqrt{14}}{5}-\frac{121}{5}\approx 1.012373203$$

Solution Steps

Factor 2016=12^{2}\times 14. Rewrite the square root of the product \sqrt{12^{2}\times 14} as the product of square roots \sqrt{12^{2}}\sqrt{14}. Take the square root of 12^{2}.

$$\frac{44-\sqrt{2017}}{44-12\sqrt{14}}$$

Rationalize the denominator of \frac{44-\sqrt{2017}}{44-12\sqrt{14}} by multiplying numerator and denominator by 44+12\sqrt{14}.

$$\frac{\left(44-\sqrt{2017}\right)\left(44+12\sqrt{14}\right)}{\left(44-12\sqrt{14}\right)\left(44+12\sqrt{14}\right)}$$

Consider \left(44-12\sqrt{14}\right)\left(44+12\sqrt{14}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(44-\sqrt{2017}\right)\left(44+12\sqrt{14}\right)}{44^{2}-\left(-12\sqrt{14}\right)^{2}}$$

Calculate 44 to the power of 2 and get 1936.

$$\frac{\left(44-\sqrt{2017}\right)\left(44+12\sqrt{14}\right)}{1936-\left(-12\sqrt{14}\right)^{2}}$$

Expand \left(-12\sqrt{14}\right)^{2}.

$$\frac{\left(44-\sqrt{2017}\right)\left(44+12\sqrt{14}\right)}{1936-\left(-12\right)^{2}\left(\sqrt{14}\right)^{2}}$$

Calculate -12 to the power of 2 and get 144.

$$\frac{\left(44-\sqrt{2017}\right)\left(44+12\sqrt{14}\right)}{1936-144\left(\sqrt{14}\right)^{2}}$$

The square of \sqrt{14} is 14.

$$\frac{\left(44-\sqrt{2017}\right)\left(44+12\sqrt{14}\right)}{1936-144\times 14}$$

Multiply 144 and 14 to get 2016.

$$\frac{\left(44-\sqrt{2017}\right)\left(44+12\sqrt{14}\right)}{1936-2016}$$

Subtract 2016 from 1936 to get -80.

$$\frac{\left(44-\sqrt{2017}\right)\left(44+12\sqrt{14}\right)}{-80}$$

Apply the distributive property by multiplying each term of 44-\sqrt{2017} by each term of 44+12\sqrt{14}.

$$\frac{1936+528\sqrt{14}-44\sqrt{2017}-12\sqrt{2017}\sqrt{14}}{-80}$$

To multiply \sqrt{2017} and \sqrt{14}, multiply the numbers under the square root.

$$\frac{1936+528\sqrt{14}-44\sqrt{2017}-12\sqrt{28238}}{-80}$$