If `10,800=2^(a)xx3^(b)xx5^( c)`, then what is the value of `a^(c)+b^(a)`?

Options

A. 97
B. 81
C. 102
D. 107

Answer

To solve the equation \( 10,800 = 2^a \times 3^b \times 5^c \), we need to factorize 10,800 into its prime factors. Here’s how we can do it step by step:

### Step 1: Factorization of 10,800
We start by dividing 10,800 by the smallest prime number, which is 2.

1. **Divide by 2:**
\[ 10,800 \div 2 = 5,400 \]
2. **Divide by 2 again:**
\[ 5,400 \div 2 = 2,700 \]
3. **Divide by 2 again:**
\[ 2,700 \div 2 = 1,350 \]
4. **Divide by 2 again:**
\[ 1,350 \div 2 = 675 \]
Now, we cannot divide by 2 anymore since 675 is odd.

### Step 2: Continue with the next prime number, which is 3
1. **Divide by 3:**
\[ 675 \div 3 = 225 \]
2. **Divide by 3 again:**
\[ 225 \div 3 = 75 \]
3. **Divide by 3 again:**
\[ 75 \div 3 = 25 \]
Now, we cannot divide by 3 anymore since 25 is not divisible by 3.

### Step 3: Continue with the next prime number, which is 5
1. **Divide by 5:**
\[ 25 \div 5 = 5 \]
2. **Divide by 5 again:**
\[ 5 \div 5 = 1 \]

### Step 4: Count the factors
Now we can summarize the factors we found:
- The number of times we divided by 2: **4 times** (so \( a = 4 \))
- The number of times we divided by 3: **3 times** (so \( b = 3 \))
- The number of times we divided by 5: **2 times** (so \( c = 2 \))

Thus, we have:
\[ 10,800 = 2^4 \times 3^3 \times 5^2 \]

### Step 5: Calculate \( a^c + b^a \)
Now we can substitute the values of \( a \), \( b \), and \( c \):
\[ a = 4, \quad b = 3, \quad c = 2 \]

Now calculate:
1. \( a^c = 4^2 = 16 \)
2. \( b^a = 3^4 = 81 \)

Finally, we add these results:
\[ a^c + b^a = 16 + 81 = 97 \]

### Final Answer
The value of \( a^c + b^a \) is \( \boxed{97} \).

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