`(6^6 + 6^6 + 6^6 + 6^6 + 6^6 + 6^6)/(3^6 + 3^6 + 3^6) div (4^6 + 4^6 + 4^6 + 4^6)/(2^6 + 2^6) = 2^n` , then the value of n is :

Options

A. `-1`
B. `0`
C. `1/2`
D. `1`

Answer

To solve the expression

\[ \frac{(6^6 + 6^6 + 6^6 + 6^6 + 6^6 + 6^6)}{(3^6 + 3^6 + 3^6)} \div \frac{(4^6 + 4^6 + 4^6 + 4^6)}{(2^6 + 2^6)} = 2^n, \]

we will simplify each part step by step.

### Step 1: Simplify the numerator of the first fraction
The numerator of the first fraction is \(6^6 + 6^6 + 6^6 + 6^6 + 6^6 + 6^6\). Since there are 6 terms of \(6^6\), we can write it as:

\[ 6 \times 6^6 = 6^1 \times 6^6 = 6^{1+6} = 6^7. \]

**Hint for Step 1:** Remember that adding the same number multiple times is equivalent to multiplying that number by the count of terms.

### Step 2: Simplify the denominator of the first fraction
The denominator of the first fraction is \(3^6 + 3^6 + 3^6\). There are 3 terms of \(3^6\), so we can write it as:

\[ 3 \times 3^6 = 3^1 \times 3^6 = 3^{1+6} = 3^7. \]

**Hint for Step 2:** Use the same principle of multiplication for addition of identical terms.

### Step 3: Write the first fraction
Now we can rewrite the first fraction:

\[ \frac{6^7}{3^7}. \]

**Hint for Step 3:** When simplifying fractions, remember that you can express them in terms of powers.

### Step 4: Simplify the first fraction
We can simplify \(\frac{6^7}{3^7}\) as follows:

\[ \left(\frac{6}{3}\right)^7 = 2^7. \]

**Hint for Step 4:** When dividing powers with the same exponent, you can divide the bases first.

### Step 5: Simplify the numerator of the second fraction
The numerator of the second fraction is \(4^6 + 4^6 + 4^6 + 4^6\). There are 4 terms of \(4^6\), so we can write it as:

\[ 4 \times 4^6 = 4^1 \times 4^6 = 4^{1+6} = 4^7. \]

**Hint for Step 5:** Again, apply the multiplication principle for identical terms.

### Step 6: Simplify the denominator of the second fraction
The denominator of the second fraction is \(2^6 + 2^6\). There are 2 terms of \(2^6\), so we can write it as:

\[ 2 \times 2^6 = 2^1 \times 2^6 = 2^{1+6} = 2^7. \]

**Hint for Step 6:** Keep using the multiplication principle for addition.

### Step 7: Write the second fraction
Now we can rewrite the second fraction:

\[ \frac{4^7}{2^7}. \]

**Hint for Step 7:** Keep the fractions in terms of their bases for easier simplification.

### Step 8: Simplify the second fraction
We can simplify \(\frac{4^7}{2^7}\) as follows:

\[ \left(\frac{4}{2}\right)^7 = 2^7. \]

**Hint for Step 8:** Again, simplify the bases before applying the exponent.

### Step 9: Combine both parts
Now we have:

\[ \frac{6^7}{3^7} \div \frac{4^7}{2^7} = 2^7 \div 2^7. \]

**Hint for Step 9:** Remember that dividing powers with the same base involves subtracting the exponents.

### Step 10: Final simplification
This simplifies to:

\[ 2^{7-7} = 2^0 = 1. \]

### Step 11: Set equal to \(2^n\)
Now we have:

\[ 1 = 2^n. \]

This implies:

\[ n = 0. \]

### Final Answer
The value of \(n\) is:

\[ \boxed{0}. \]