Simplify 3/(2x)*(x^3)/3

Solution:

Step 1:

Combine the terms. $\frac{3}{2x} \cdot \frac{x^3}{3}$

Step 2:

Eliminate the common factor of $3$.

Step 2.1:

Remove the common factor. $\frac{\cancel{3} x^{3}}{2x \cdot \cancel{3}}$

Step 2.2:

Simplify the expression. $\frac{x^{3}}{2x}$

Step 3:

Reduce the terms with $x$.

Step 3.1:

Extract $x$ from $x^3$. $\frac{x \cdot x^{2}}{2x}$

Step 3.2:

Eliminate the common $x$ terms.

Step 3.2.1:

Factor out $x$ from $2x$. $\frac{x \cdot x^{2}}{x \cdot 2}$

Step 3.2.2:

Remove the common $x$. $\frac{\cancel{x} \cdot x^{2}}{\cancel{x} \cdot 2}$

Step 3.2.3:

Finalize the expression. $\frac{x^{2}}{2}$

Knowledge Notes:

To simplify the expression $\frac{3}{2x} \cdot \frac{x^3}{3}$, we need to follow a systematic approach:

1. Combining Fractions: When two fractions are multiplied, we multiply the numerators together and the denominators together.

2. Canceling Common Factors: If the numerator and denominator share a common factor, they can be canceled out. This is based on the property that $\frac{a}{a} = 1$ for any non-zero $a$.

3. Reducing Powers: When we have the same base in both the numerator and denominator, we can subtract the exponents if the base is raised to a power in both. This is due to the property $x^m / x^n = x^{m-n}$, where $m$ and $n$ are integers and $x$ is not equal to zero.

4. Simplification: The final step is to rewrite the expression in its simplest form by ensuring that there are no common factors left and that the expression is as compact as possible.

By applying these principles, we can simplify the given expression step by step, ensuring that we do not skip any important simplification processes.