If `2160 = 2^(a) *3^b* 5^( c)`, then find the value of `3^(a) xx 2^(-b) xx 5^(-c)`

Options

A. `81/40`
B. 0
C. `1/2`
D. `37/39`

Explanation

To solve the equation \( 2160 = 2^{a} \cdot 3^{b} \cdot 5^{c} \) and find the value of \( 3^{a} \cdot 2^{-b} \cdot 5^{-c} \), we will first factorize 2160 into its prime factors.

### Step 1: Factorize 2160
We start by dividing 2160 by the smallest prime numbers until we reach 1.

1. Divide by 2:
\[ 2160 \div 2 = 1080 \]
2. Divide by 2 again:
\[ 1080 \div 2 = 540 \]
3. Divide by 2 again:
\[ 540 \div 2 = 270 \]
4. Divide by 2 again:
\[ 270 \div 2 = 135 \]
5. Now, divide by 3:
\[ 135 \div 3 = 45 \]
6. Divide by 3 again:
\[ 45 \div 3 = 15 \]
7. Divide by 3 again:
\[ 15 \div 3 = 5 \]
8. Finally, divide by 5:
\[ 5 \div 5 = 1 \]

### Step 2: Count the factors
From the above divisions, we can count the number of times we divided by each prime factor:
- The factor of 2 appears 4 times.
- The factor of 3 appears 3 times.
- The factor of 5 appears 1 time.

Thus, we can express 2160 as:
\[ 2160 = 2^4 \cdot 3^3 \cdot 5^1 \]

### Step 3: Identify the values of a, b, and c
From our factorization, we can identify:
- \( a = 4 \)
- \( b = 3 \)
- \( c = 1 \)

### Step 4: Substitute into the expression
Now we need to find the value of \( 3^{a} \cdot 2^{-b} \cdot 5^{-c} \):
\[ 3^{a} \cdot 2^{-b} \cdot 5^{-c} = 3^{4} \cdot 2^{-3} \cdot 5^{-1} \]

### Step 5: Calculate each term
1. Calculate \( 3^{4} \):
\[ 3^{4} = 81 \]
2. Calculate \( 2^{-3} \):
\[ 2^{-3} = \frac{1}{2^3} = \frac{1}{8} \]
3. Calculate \( 5^{-1} \):
\[ 5^{-1} = \frac{1}{5} \]

### Step 6: Combine the results
Now we combine these results:
\[ 3^{4} \cdot 2^{-3} \cdot 5^{-1} = 81 \cdot \frac{1}{8} \cdot \frac{1}{5} \]

Calculating this step by step:
1. First, calculate \( 81 \cdot \frac{1}{8} \):
\[ \frac{81}{8} \]
2. Now multiply by \( \frac{1}{5} \):
\[ \frac{81}{8} \cdot \frac{1}{5} = \frac{81}{40} \]

### Final Answer
Thus, the value of \( 3^{a} \cdot 2^{-b} \cdot 5^{-c} \) is:
\[ \frac{81}{40} \]

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