24x3           30
------  x  -----------
50x              8x

To solve the given expression, we'll start by simplifying the multiplication and fractions step by step. The expression is:

\[
\frac{24x^3}{50x} \times \frac{30}{8x}
\]

### Step 1: Simplify Each Fraction

First, simplify each fraction separately.

1. **Simplify** \(\frac{24x^3}{50x}\):

\[
\frac{24x^3}{50x} = \frac{24}{50} \times \frac{x^3}{x} = \frac{12}{25} \times x^2 = \frac{12x^2}{25}
\]

2. **Simplify** \(\frac{30}{8x}\):

\[
\frac{30}{8x} = \frac{15}{4x}
\]

### Step 2: Multiply the Two Simplified Fractions

Now, multiply the two simplified fractions:

\[
\frac{12x^2}{25} \times \frac{15}{4x}
\]

Multiply the numerators and denominators:

\[
= \frac{12x^2 \cdot 15}{25 \cdot 4x}
\]

### Step 3: Perform Multiplication and Simplify

Calculate the multiplication in the numerator and the denominator:

\[
= \frac{180x^2}{100x}
\]

Now, simplify the fraction by dividing the numerator and denominator by their greatest common divisor (GCD):

\[
= \frac{180x^2 \div 20}{100x \div 20} = \frac{9x}{5}
\]

### Final Answer

\[
\frac{9x}{5}
\]

The simplified expression is \(\frac{9x}{5}\).

Would you like more details or have any questions?

Here are some related questions to explore:
1. What is the importance of simplifying fractions in mathematical expressions?
2. How do you find the greatest common divisor (GCD) of two numbers?
3. Can this expression be simplified further if additional conditions are given for \(x\)?
4. How do you handle expressions with multiple variables in fractions?
5. What are the steps to factor polynomials in fraction expressions?

**Tip:** Always check for common factors in both the numerator and denominator to simplify fractions effectively!

Learn how to simplify and multiply algebraic fractions using the expression 24x^3 / 50x * 30 / 8x. This step-by-step guide covers key algebraic concepts, such as fraction simplification, properties of exponents, and finding the greatest common divisor (GCD). Suitable for students in Grades 8-10.

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