Solve frac{2sqrt{45}x}{3sqrt{10}}

$$\frac { 2 \sqrt { 45 } x } { 3 \sqrt { 10 } }$$

Evaluate

$$\sqrt{2}x$$

Solution Steps

Factor 45=3^{2}\times 5. Rewrite the square root of the product \sqrt{3^{2}\times 5} as the product of square roots \sqrt{3^{2}}\sqrt{5}. Take the square root of 3^{2}.

$$\frac{2\times 3\sqrt{5}x}{3\sqrt{10}}$$

Multiply 2 and 3 to get 6.

$$\frac{6\sqrt{5}x}{3\sqrt{10}}$$

Cancel out 3 in both numerator and denominator.

$$\frac{2\sqrt{5}x}{\sqrt{10}}$$

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

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

The square of \sqrt{10} is 10.

$$\frac{2\sqrt{5}x\sqrt{10}}{10}$$

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}.

$$\frac{2\sqrt{5}x\sqrt{5}\sqrt{2}}{10}$$

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

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

Multiply 2 and 5 to get 10.

$$\frac{10x\sqrt{2}}{10}$$

Cancel out 10 and 10.

$$x\sqrt{2}$$

Differentiate w.r.t. x

$$\sqrt{2} = 1.414213562$$