Solve sqrt{208/11}n-sqrt{21/2}

$$\sqrt { \frac { 208 } { 11 } } n - \sqrt { \frac { 21 } { 2 } }$$

Evaluate

$$\frac{4\sqrt{143}n}{11}-\frac{\sqrt{42}}{2}$$

Short Solution Steps

Rewrite the square root of the division \sqrt{\frac{208}{11}} as the division of square roots \frac{\sqrt{208}}{\sqrt{11}}.

$$\frac{\sqrt{208}}{\sqrt{11}}n-\sqrt{\frac{21}{2}}$$

Factor 208=4^{2}\times 13. Rewrite the square root of the product \sqrt{4^{2}\times 13} as the product of square roots \sqrt{4^{2}}\sqrt{13}. Take the square root of 4^{2}.

$$\frac{4\sqrt{13}}{\sqrt{11}}n-\sqrt{\frac{21}{2}}$$

Rationalize the denominator of \frac{4\sqrt{13}}{\sqrt{11}} by multiplying numerator and denominator by \sqrt{11}.

$$\frac{4\sqrt{13}\sqrt{11}}{\left(\sqrt{11}\right)^{2}}n-\sqrt{\frac{21}{2}}$$

The square of \sqrt{11} is 11.

$$\frac{4\sqrt{13}\sqrt{11}}{11}n-\sqrt{\frac{21}{2}}$$

To multiply \sqrt{13} and \sqrt{11}, multiply the numbers under the square root.

$$\frac{4\sqrt{143}}{11}n-\sqrt{\frac{21}{2}}$$

Express \frac{4\sqrt{143}}{11}n as a single fraction.

$$\frac{4\sqrt{143}n}{11}-\sqrt{\frac{21}{2}}$$

Rewrite the square root of the division \sqrt{\frac{21}{2}} as the division of square roots \frac{\sqrt{21}}{\sqrt{2}}.

$$\frac{4\sqrt{143}n}{11}-\frac{\sqrt{21}}{\sqrt{2}}$$

Rationalize the denominator of \frac{\sqrt{21}}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{4\sqrt{143}n}{11}-\frac{\sqrt{21}\sqrt{2}}{\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{4\sqrt{143}n}{11}-\frac{\sqrt{21}\sqrt{2}}{2}$$

To multiply \sqrt{21} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{4\sqrt{143}n}{11}-\frac{\sqrt{42}}{2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 11 and 2 is 22. Multiply \frac{4\sqrt{143}n}{11} times \frac{2}{2}. Multiply \frac{\sqrt{42}}{2} times \frac{11}{11}.

$$\frac{2\times 4\sqrt{143}n}{22}-\frac{11\sqrt{42}}{22}$$

Since \frac{2\times 4\sqrt{143}n}{22} and \frac{11\sqrt{42}}{22} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2\times 4\sqrt{143}n-11\sqrt{42}}{22}$$

Do the multiplications in 2\times 4\sqrt{143}n-11\sqrt{42}.

$$\frac{8n\sqrt{143}-11\sqrt{42}}{22}$$

Differentiate w.r.t. n

$$\frac{4 \sqrt{143}}{11} = 4.348458452036872$$