`(27)^3`*`3^4`/`(81)^2`=`3^?`

Options

A. 2
B. 5
C. 4
D. 3

Answer

To solve the equation \( (27)^3 \cdot (3^4) / (81)^2 = 3^? \), we will simplify the left-hand side step by step.

### Step 1: Rewrite the numbers as powers of 3
First, we express 27 and 81 in terms of base 3:
- \( 27 = 3^3 \)
- \( 81 = 3^4 \)

### Step 2: Substitute the powers into the equation
Now we can rewrite the equation:
\[ (3^3)^3 \cdot (3^4) / (3^4)^2 = 3^? \]

### Step 3: Apply the power of a power property
Using the property \( (a^m)^n = a^{m \cdot n} \), we simplify:
\[ 3^{3 \cdot 3} \cdot 3^4 / 3^{4 \cdot 2} = 3^? \]
This simplifies to:
\[ 3^9 \cdot 3^4 / 3^8 = 3^? \]

### Step 4: Combine the powers in the numerator
Using the property \( a^m \cdot a^n = a^{m+n} \):
\[ 3^{9 + 4} / 3^8 = 3^? \]
This simplifies to:
\[ 3^{13} / 3^8 = 3^? \]

### Step 5: Apply the division property
Using the property \( \frac{a^m}{a^n} = a^{m-n} \):
\[ 3^{13 - 8} = 3^? \]
This simplifies to:
\[ 3^5 = 3^? \]

### Step 6: Equate the exponents
Since the bases are the same, we can equate the exponents:
\[ ? = 5 \]

### Conclusion
Thus, the value of \( ? \) is \( 5 \).