The values of m, n respectively, if `108 = 2^m xx 3^3 xx 5^n`, are:

Options

A. `2,0 `
B. `3,1`
C. ` 0,1 `
D. `2,2`

Answer

To find the values of \( m \) and \( n \) in the equation \( 108 = 2^m \times 3^3 \times 5^n \), we will follow these steps:

### Step 1: Factorize 108
First, we need to factor 108 into its prime factors.

\[ 108 = 2 \times 54 \]
\[ 54 = 2 \times 27 \]
\[ 27 = 3 \times 9 \]
\[ 9 = 3 \times 3 \]

So, we can express 108 as:
\[ 108 = 2^2 \times 3^3 \]

### Step 2: Write the equation
Now, we can write the equation based on the prime factorization we found:
\[ 108 = 2^2 \times 3^3 \times 5^0 \]

### Step 3: Compare with the given equation
We have:
\[ 2^m \times 3^3 \times 5^n = 2^2 \times 3^3 \times 5^0 \]

### Step 4: Equate the powers
Now we can equate the powers of the corresponding bases:
- For base 2: \( m = 2 \)
- For base 3: The power is already 3, so it confirms our equation.
- For base 5: \( n = 0 \)

### Final Values
Thus, the values of \( m \) and \( n \) are:
\[ m = 2, \quad n = 0 \]

### Answer
The values of \( m \) and \( n \) respectively are \( 2 \) and \( 0 \).

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