2/5x3/6x4/7x5/8x6/9x7/10x8/11x9/12x10/13x11/14x12/15x13/16x14/17x15/18x16/19x17/20x18/21x19/22x20/23x21/24

To simplify the given expression, we need to multiply all the fractions together:

\[
\frac{2}{5} \times \frac{3}{6} \times \frac{4}{7} \times \frac{5}{8} \times \frac{6}{9} \times \frac{7}{10} \times \frac{8}{11} \times \frac{9}{12} \times \frac{10}{13} \times \frac{11}{14} \times \frac{12}{15} \times \frac{13}{16} \times \frac{14}{17} \times \frac{15}{18} \times \frac{16}{19} \times \frac{17}{20} \times \frac{18}{21} \times \frac{19}{22} \times \frac{20}{23} \times \frac{21}{24}
\]

First, let's write the product of numerators and the product of denominators separately:

Numerators: \(2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9 \times 10 \times 11 \times 12 \times 13 \times 14 \times 15 \times 16 \times 17 \times 18 \times 19 \times 20 \times 21\)

Denominators: \(5 \times 6 \times 7 \times 8 \times 9 \times 10 \times 11 \times 12 \times 13 \times 14 \times 15 \times 16 \times 17 \times 18 \times 19 \times 20 \times 21 \times 22 \times 23 \times 24\)

Now, let's cancel out the common factors from the numerators and denominators:

\[
\frac{2}{5} \times \frac{3}{6} \times \frac{4}{7} \times \frac{5}{8} \times \frac{6}{9} \times \frac{7}{10} \times \frac{8}{11} \times \frac{9}{12} \times \frac{10}{13} \times \frac{11}{14} \times \frac{12}{15} \times \frac{13}{16} \times \frac{14}{17} \times \frac{15}{18} \times \frac{16}{19} \times \frac{17}{20} \times \frac{18}{21} \times \frac{19}{22} \times \frac{20}{23} \times \frac{21}{24}
\]

\[
= \frac{2}{5} \times \frac{1}{2} \times \frac{4}{7} \times \frac{1}{8} \times \frac{2}{3} \times \frac{1}{10} \times \frac{8}{11} \times \frac{3}{4} \times \frac{10}{13} \times \frac{11}{14} \times \frac{4}{5} \times \frac{13}{16} \times \frac{14}{17} \times \frac{5}{6} \times \frac{16}{19} \times \frac{17}{20} \times \frac{6}{7} \times \frac{19}{22} \times \frac{20}{23} \times \frac{7}{8}
\]

\[
= \frac{1 \cdot 4 \cdot 1 \cdot 2 \cdot 1 \cdot 8 \cdot 3 \cdot 10 \cdot 11 \cdot 4 \cdot 13 \cdot 14 \cdot 5 \cdot 16 \cdot 17 \cdot 6 \cdot 19 \cdot 20 \cdot 7}{5 \cdot 7 \cdot 8 \cdot 3 \cdot 10 \cdot 11 \cdot 4 \cdot 13 \cdot 14 \cdot 5 \cdot 16 \cdot 17 \cdot 6 \cdot 19 \cdot 20 \cdot 7 \cdot 22 \cdot 23 \cdot 8}
\]

Cancel out all the common terms:

\[
= \frac{1 \cdot 1}{5 \cdot 22 \cdot 23}
\]

The numerator is 1, and the remaining product of the denominators is:

\[
5 \times 22 \times 23 = 2530
\]

Therefore, the simplified form of the given expression is:

\[
\frac{1}{2530}
\]

Would you like more details or have any questions?

Here are some related questions:

1. How do you simplify a complex fraction?
2. What are the rules for canceling out terms in fractions?
3. How do you handle products of fractions with large numerators and denominators?
4. Can you provide an example of simplifying a different sequence of fractions?
5. How do you approach simplifying an expression with mixed numbers?

**Tip:** When simplifying fractions, always look for common factors between the numerator and the denominator to cancel them out efficiently.

Learn how to simplify the multiplication of complex fractions: 2/5 x 3/6 x 4/7... up to 23/21 x 24. This step-by-step guide covers key concepts in fraction multiplication and simplification for students in Grades 8-10.

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