Solve Tfrac{sqrt{320}-sqrt{80}}{sqrt{40}}

$$T \frac { \sqrt { 320 } - \sqrt { 80 } } { \sqrt { 40 } }$$

Evaluate

$$\sqrt{2}T$$

Solution Steps

Factor 320=8^{2}\times 5. Rewrite the square root of the product \sqrt{8^{2}\times 5} as the product of square roots \sqrt{8^{2}}\sqrt{5}. Take the square root of 8^{2}.

$$T\times \frac{8\sqrt{5}-\sqrt{80}}{\sqrt{40}}$$

Factor 80=4^{2}\times 5. Rewrite the square root of the product \sqrt{4^{2}\times 5} as the product of square roots \sqrt{4^{2}}\sqrt{5}. Take the square root of 4^{2}.

$$T\times \frac{8\sqrt{5}-4\sqrt{5}}{\sqrt{40}}$$

Combine 8\sqrt{5} and -4\sqrt{5} to get 4\sqrt{5}.

$$T\times \frac{4\sqrt{5}}{\sqrt{40}}$$

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}.

$$T\times \frac{4\sqrt{5}}{2\sqrt{10}}$$

Cancel out 2 in both numerator and denominator.

$$T\times \frac{2\sqrt{5}}{\sqrt{10}}$$

Rationalize the denominator of \frac{2\sqrt{5}}{\sqrt{10}} by multiplying numerator and denominator by \sqrt{10}.

$$T\times \frac{2\sqrt{5}\sqrt{10}}{\left(\sqrt{10}\right)^{2}}$$

The square of \sqrt{10} is 10.

$$T\times \frac{2\sqrt{5}\sqrt{10}}{10}$$

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

$$T\times \frac{2\sqrt{5}\sqrt{5}\sqrt{2}}{10}$$

Multiply \sqrt{5} and \sqrt{5} to get 5.

$$T\times \frac{2\times 5\sqrt{2}}{10}$$

Multiply 2 and 5 to get 10.

$$T\times \frac{10\sqrt{2}}{10}$$

Cancel out 10 and 10.

$$T\sqrt{2}$$

Differentiate w.r.t. T

$$\sqrt{2} = 1.414213562$$