Solve sqrt{5}x(sqrt{15}-4sqrt{3/5})

$$\sqrt { 5 } x ( \sqrt { 15 } - 4 \sqrt { \frac { 3 } { 5 } } )$$

Evaluate

$$\sqrt{3}x$$

Short Solution Steps

Rewrite the square root of the division \sqrt{\frac{3}{5}} as the division of square roots \frac{\sqrt{3}}{\sqrt{5}}.

$$\sqrt{5}x\left(\sqrt{15}-4\times \frac{\sqrt{3}}{\sqrt{5}}\right)$$

Rationalize the denominator of \frac{\sqrt{3}}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\sqrt{5}x\left(\sqrt{15}-4\times \frac{\sqrt{3}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}\right)$$

The square of \sqrt{5} is 5.

$$\sqrt{5}x\left(\sqrt{15}-4\times \frac{\sqrt{3}\sqrt{5}}{5}\right)$$

To multiply \sqrt{3} and \sqrt{5}, multiply the numbers under the square root.

$$\sqrt{5}x\left(\sqrt{15}-4\times \frac{\sqrt{15}}{5}\right)$$

Express -4\times \frac{\sqrt{15}}{5} as a single fraction.

$$\sqrt{5}x\left(\sqrt{15}+\frac{-4\sqrt{15}}{5}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply \sqrt{15} times \frac{5}{5}.

$$\sqrt{5}x\left(\frac{5\sqrt{15}}{5}+\frac{-4\sqrt{15}}{5}\right)$$

Since \frac{5\sqrt{15}}{5} and \frac{-4\sqrt{15}}{5} have the same denominator, add them by adding their numerators.

$$\sqrt{5}x\times \frac{5\sqrt{15}-4\sqrt{15}}{5}$$

Do the calculations in 5\sqrt{15}-4\sqrt{15}.

$$\sqrt{5}x\times \frac{\sqrt{15}}{5}$$

Express \sqrt{5}\times \frac{\sqrt{15}}{5} as a single fraction.

$$\frac{\sqrt{5}\sqrt{15}}{5}x$$

Factor 15=5\times 3. Rewrite the square root of the product \sqrt{5\times 3} as the product of square roots \sqrt{5}\sqrt{3}.

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

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

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

Cancel out 5 and 5.

$$\sqrt{3}x$$

Differentiate w.r.t. x

$$\sqrt{3} = 1.732050808$$