To find the value of the expression \(3^a \times 2^{-b} \times 5^{-c}\), we first need to determine the numerical values of \(a\), \(b\), and \(c\). These values are derived from the given equation \(2160 = 2^a \times 3^b \times 5^c\). This requires performing the prime factorization of the number 2160.

Prime Factorization of 2160

The prime factorization of a number is the process of breaking it down into its prime number components. For the number 2160, we will systematically divide it by the smallest possible prime numbers (2, 3, 5, etc.) until the result is 1. The steps are shown in the table below:

From the detailed factorization process, we can observe that 2160 is composed of four factors of 2, three factors of 3, and one factor of 5. Therefore, the prime factorization of 2160 can be expressed in exponential form as:

\[ 2160 = 2^4 \times 3^3 \times 5^1 \]

Now, we compare this prime factorization with the given equation \(2160 = 2^a \times 3^b \times 5^c\). By matching the exponents of the corresponding prime bases, we can determine the values for \(a\), \(b\), and \(c\):

• The exponent of base 2 in \(2^4\) is 4, so \(a = 4\).
• The exponent of base 3 in \(3^3\) is 3, so \(b = 3\).
• The exponent of base 5 in \(5^1\) is 1, so \(c = 1\).

Evaluating the Expression \(3^a \times 2^{-b} \times 5^{-c}\)

With the values of \(a=4\), \(b=3\), and \(c=1\) successfully identified, we can now substitute these values into the expression \(3^a \times 2^{-b} \times 5^{-c}\).

Substituting the values, the expression becomes:

\[ 3^4 \times 2^{-3} \times 5^{-1} \]

To simplify terms with negative exponents, we apply the fundamental property of exponents: \(x^{-n} = \frac{1}{x^n}\). Let's apply this rule to \(2^{-3}\) and \(5^{-1}\):

For \(2^{-3}\):

\[ 2^{-3} = \frac{1}{2^3} = \frac{1}{2 \times 2 \times 2} = \frac{1}{8} \]

For \(5^{-1}\):

\[ 5^{-1} = \frac{1}{5^1} = \frac{1}{5} \]

Next, we calculate the value of \(3^4\):

\[ 3^4 = 3 \times 3 \times 3 \times 3 = 81 \]

Now, we substitute all these calculated individual values back into the expression:

\[ 81 \times \frac{1}{8} \times \frac{1}{5} \]

Finally, we perform the multiplication of these terms:

\[ \frac{81 \times 1 \times 1}{1 \times 8 \times 5} = \frac{81}{40} \]

Therefore, the value of the expression \(3^a \times 2^{-b} \times 5^{-c}\) is \(\frac{81}{40}\).