Question

Simplify the expression. $$\left(\frac{x^{2}}{5} \cdot \frac{x+2}{2}\right) \div \frac{x}{30}$$

Short answer

The simplified form of the expression \(\left(\frac{x^{2}}{5} \cdot \frac{x+2}{2}\right) \div \frac{x}{30}\) is \(3x^{2}+3x\)

Step-by-step solution

Handle inner operation

Starting with the multiplication part inside the brackets first, distribute the \(\frac{x^{2}}{5}\) across \(\frac{x+2}{2}\) to get \(\frac{x^{3}}{10}+\frac{x^{2}}{10}\)

Handle the division operation

Now we divide the result from the step 1 by \(\frac{x}{30}\), that is, \((\frac{x^{3}}{10}+\frac{x^{2}}{10}) \div \(\frac{x}{30}\) \). By the rule of division of fractions, this can be changed to multiplication, which leads to \((\frac{x^{3}}{10}+\frac{x^{2}}{10}) \cdot \frac{30}{x} \).

Distribute the multiplied term

In the expression obtained from step 2, distribute the \(\frac{30}{x}\) to \( \frac{x^{3}}{10}+\frac{x^{2}}{10} \). This yields \(\frac{3x^{2}}{1} + \frac{3x}{1}\), which simplifies to \(3x^{2}+3x\)