Verified Answer :

  • Prime factorize each numerator and denominator: \(21=3\times7\), \(81=3^4\), \(108=2^2\times3^3\), and \(56=2^3\times7\).
  • Substitute these into the expression: \(\frac{3\times7}{3^4} \times \frac{2^2\times3^3}{2^3\times7}\).
  • Simplify by canceling common factors \(3^4\) and \(7\), leaving \(\frac{2^2}{2^3} = \frac{1}{2}\).
  • Final result: \(\boxed{\frac{1}{2}}\).

Explanation

  1. Analyzing and Factoring the Fractions

The objective is to compute the value of the expression \(\frac{21}{81} \times \frac{108}{56}\). To do this efficiently, we can simplify each fraction individually before performing the multiplication or cross-cancel common factors between the numerators and the denominators. Let's break down each number into its prime factors to make the simplification process clear.
The numerators are \(21\) and \(108\), and the denominators are \(81\) and \(56\).

  • \(21 = 3 \times 7\)
  • \(81 = 9 \times 9 = 3^2 \times 3^2 = 3^4\)
  • \(108 = 2 \times 54 = 2 \times 2 \times 27 = 2^2 \times 3^3\)
  • \(56 = 8 \times 7 = 2^3 \times 7\)
  1. Simplifying the Expression

Now we can rewrite the product of the fractions using these factorizations:
\[ \frac{21}{81} \times \frac{108}{56} = \frac{3 \times 7}{3^4} \times \frac{2^2 \times 3^3}{2^3 \times 7} \]
Combining these into a single fraction gives:
\[ \frac{3 \times 7 \times 2^2 \times 3^3}{3^4 \times 2^3 \times 7} \]
Next, we simplify the expression by canceling the common terms found in both the numerator and the denominator:

  • The factor \(7\) appears in both, so it cancels out.
  • The factor \(3^1 \times 3^3 = 3^4\) in the numerator cancels with the \(3^4\) in the denominator.
  • We are left with \(\frac{2^2}{2^3}\) in the expression.

\[ \frac{2^2}{2^3} = \frac{1}{2^1} = \frac{1}{2} \]

  1. Final Calculation and Result

After simplifying the fractions by canceling out the common prime factors, the final result is \(\frac{1}{2}\). Thus, \(\frac{21}{81} \times \frac{108}{56} = \boxed{\frac{1}{2}}\).

Examples

Understanding how to simplify fractions through prime factorization is essential for scaling recipes, calculating discounts, or managing resources in everyday life. For instance, if a recipe calls for \(\frac{21}{81}\) of a cup of an ingredient and you want to scale the recipe by \(\frac{108}{56}\), knowing how to perform these operations ensures you use the correct amounts without having to deal with unwieldy, large numbers. This skill makes quantitative tasks more manageable and less prone to calculation errors.