What is value of `[(3 × 3 × 3 × 3 × 3 × 3)^6 ÷ (3 × 3 × 3 × 3)^7× 3^4]?`

Options

A. a. `27^9`
B. b. `27^4`
C. c. `31^5`
D. d. `91^0`

Answer

To solve the expression \(\left[\frac{(3 \times 3 \times 3 \times 3 \times 3 \times 3)^6}{(3 \times 3 \times 3 \times 3)^7 \times 3^4}\right]\), we can simplify it step by step.

### Step 1: Rewrite the expression using exponents
The expression can be rewritten using exponents:
\[ (3^6)^6 \div (3^4)^7 \times 3^4 \]
This simplifies to:
\[ 3^{6 \times 6} \div 3^{4 \times 7} \times 3^4 \]

### Step 2: Calculate the exponents
Now we calculate the exponents:
\[ 3^{36} \div 3^{28} \times 3^4 \]

### Step 3: Apply the division rule of exponents
Using the rule of exponents \(\frac{a^m}{a^n} = a^{m-n}\), we can simplify:
\[ 3^{36 - 28} \times 3^4 \]
This simplifies to:
\[ 3^8 \times 3^4 \]

### Step 4: Apply the multiplication rule of exponents
Using the rule of exponents \(a^m \times a^n = a^{m+n}\):
\[ 3^{8 + 4} = 3^{12} \]

### Step 5: Calculate \(3^{12}\)
Now we need to calculate \(3^{12}\). This can be expressed as:
\[ (3^3)^4 = 27^4 \]

### Final Answer
Thus, the value of the original expression is:
\[ 27^4 \]