Solve sqrt{2015/2016}divsqrt{2016/2017}

$$\sqrt{ \frac{ 2015 }{ 2016 } } \div \sqrt{ \frac{ 2016 }{ 2017 } }$$

Evaluate

$$\frac{\sqrt{4064255}}{2016}\approx 0.999999877$$

Solution Steps

Rewrite the square root of the division \sqrt{\frac{2015}{2016}} as the division of square roots \frac{\sqrt{2015}}{\sqrt{2016}}.

$$\frac{\frac{\sqrt{2015}}{\sqrt{2016}}}{\sqrt{\frac{2016}{2017}}}$$

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}.

$$\frac{\frac{\sqrt{2015}}{12\sqrt{14}}}{\sqrt{\frac{2016}{2017}}}$$

Rationalize the denominator of \frac{\sqrt{2015}}{12\sqrt{14}} by multiplying numerator and denominator by \sqrt{14}.

$$\frac{\frac{\sqrt{2015}\sqrt{14}}{12\left(\sqrt{14}\right)^{2}}}{\sqrt{\frac{2016}{2017}}}$$

The square of \sqrt{14} is 14.

$$\frac{\frac{\sqrt{2015}\sqrt{14}}{12\times 14}}{\sqrt{\frac{2016}{2017}}}$$

To multiply \sqrt{2015} and \sqrt{14}, multiply the numbers under the square root.

$$\frac{\frac{\sqrt{28210}}{12\times 14}}{\sqrt{\frac{2016}{2017}}}$$

Multiply 12 and 14 to get 168.

$$\frac{\frac{\sqrt{28210}}{168}}{\sqrt{\frac{2016}{2017}}}$$

Rewrite the square root of the division \sqrt{\frac{2016}{2017}} as the division of square roots \frac{\sqrt{2016}}{\sqrt{2017}}.

$$\frac{\frac{\sqrt{28210}}{168}}{\frac{\sqrt{2016}}{\sqrt{2017}}}$$

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}.

$$\frac{\frac{\sqrt{28210}}{168}}{\frac{12\sqrt{14}}{\sqrt{2017}}}$$

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

$$\frac{\frac{\sqrt{28210}}{168}}{\frac{12\sqrt{14}\sqrt{2017}}{\left(\sqrt{2017}\right)^{2}}}$$

The square of \sqrt{2017} is 2017.

$$\frac{\frac{\sqrt{28210}}{168}}{\frac{12\sqrt{14}\sqrt{2017}}{2017}}$$

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

$$\frac{\frac{\sqrt{28210}}{168}}{\frac{12\sqrt{28238}}{2017}}$$

Divide \frac{\sqrt{28210}}{168} by \frac{12\sqrt{28238}}{2017} by multiplying \frac{\sqrt{28210}}{168} by the reciprocal of \frac{12\sqrt{28238}}{2017}.

$$\frac{\sqrt{28210}\times 2017}{168\times 12\sqrt{28238}}$$

Rationalize the denominator of \frac{\sqrt{28210}\times 2017}{168\times 12\sqrt{28238}} by multiplying numerator and denominator by \sqrt{28238}.

$$\frac{\sqrt{28210}\times 2017\sqrt{28238}}{168\times 12\left(\sqrt{28238}\right)^{2}}$$

The square of \sqrt{28238} is 28238.

$$\frac{\sqrt{28210}\times 2017\sqrt{28238}}{168\times 12\times 28238}$$

To multiply \sqrt{28210} and \sqrt{28238}, multiply the numbers under the square root.

$$\frac{\sqrt{796593980}\times 2017}{168\times 12\times 28238}$$

Multiply 168 and 12 to get 2016.

$$\frac{\sqrt{796593980}\times 2017}{2016\times 28238}$$

Multiply 2016 and 28238 to get 56927808.

$$\frac{\sqrt{796593980}\times 2017}{56927808}$$

Factor 796593980=14^{2}\times 4064255. Rewrite the square root of the product \sqrt{14^{2}\times 4064255} as the product of square roots \sqrt{14^{2}}\sqrt{4064255}. Take the square root of 14^{2}.

$$\frac{14\sqrt{4064255}\times 2017}{56927808}$$

Multiply 14 and 2017 to get 28238.

$$\frac{28238\sqrt{4064255}}{56927808}$$

Divide 28238\sqrt{4064255} by 56927808 to get \frac{1}{2016}\sqrt{4064255}.

$$\frac{1}{2016}\sqrt{4064255}$$