Simplify : `(2 xx (3)^4 xx (2)^5)/(9 xx 4)`

Options

A. 0
B. 72
C. 18
D. 144

Answer

To simplify the expression \((2 \times (3)^4 \times (2)^5)/(9 \times 4)\), we can follow these steps:

### Step 1: Rewrite the expression
The expression can be rewritten as:
\[ \frac{2^1 \times 3^4 \times 2^5}{9 \times 4} \]

### Step 2: Combine the powers of 2
Using the property of exponents \(a^m \times a^n = a^{m+n}\), we can combine the powers of 2:
\[ 2^1 \times 2^5 = 2^{1+5} = 2^6 \]
So the expression now looks like:
\[ \frac{2^6 \times 3^4}{9 \times 4} \]

### Step 3: Rewrite the denominator
Next, we can rewrite the denominator \(9\) and \(4\) in terms of their prime factors:
\[ 9 = 3^2 \quad \text{and} \quad 4 = 2^2 \]
Thus, the denominator becomes:
\[ 9 \times 4 = 3^2 \times 2^2 \]

### Step 4: Substitute back into the expression
Now we substitute the denominator back into the expression:
\[ \frac{2^6 \times 3^4}{3^2 \times 2^2} \]

### Step 5: Simplify the expression
Using the property of exponents for division \(a^m / a^n = a^{m-n}\), we can simplify both the powers of 2 and the powers of 3:
\[ \frac{2^6}{2^2} = 2^{6-2} = 2^4 \]
\[ \frac{3^4}{3^2} = 3^{4-2} = 3^2 \]
So the expression simplifies to:
\[ 2^4 \times 3^2 \]

### Step 6: Calculate the values
Now we can calculate \(2^4\) and \(3^2\):
\[ 2^4 = 16 \quad \text{and} \quad 3^2 = 9 \]
Thus, we have:
\[ 2^4 \times 3^2 = 16 \times 9 \]

### Step 7: Final multiplication
Now we multiply \(16\) and \(9\):
\[ 16 \times 9 = 144 \]

### Final Answer
The simplified expression is:
\[ \boxed{144} \]