`(frac(1) (216))^(frac(2)(3)) div (frac(1)(27))^(frac(4)(3))` is equal to

Options

A. `frac(3)(4)`
B. `frac(2)(3)`
C. `frac(9)(4)`
D. `frac(1)(8)`

Answer

To solve the expression \((\frac{1}{216})^{\frac{2}{3}} \div (\frac{1}{27})^{\frac{4}{3}}\), we will follow these steps:

### Step 1: Factor the numbers
First, we need to factor the numbers 216 and 27 into their prime factors.

- **For 216**:
- \(216\) is even, so we divide by \(2\):
- \(216 \div 2 = 108\)
- \(108 \div 2 = 54\)
- \(54 \div 2 = 27\)
- Now, \(27\) is \(3^3\) (since \(3 \times 3 \times 3 = 27\)).
- Therefore, \(216 = 2^3 \times 3^3\).

- **For 27**:
- \(27 = 3^3\).

### Step 2: Rewrite the expression using prime factors
Now we can rewrite the original expression using these factors:
\[ \left(\frac{1}{2^3 \times 3^3}\right)^{\frac{2}{3}} \div \left(\frac{1}{3^3}\right)^{\frac{4}{3}} \]

### Step 3: Apply the exponent rules
Using the exponent rules, we can simplify each part:
\[ \left(\frac{1}{2^3 \times 3^3}\right)^{\frac{2}{3}} = \frac{1^{\frac{2}{3}}}{(2^3 \times 3^3)^{\frac{2}{3}}} = \frac{1}{2^{2} \times 3^{2}} = \frac{1}{4 \times 9} = \frac{1}{36} \]

For the second part:
\[ \left(\frac{1}{3^3}\right)^{\frac{4}{3}} = \frac{1^{\frac{4}{3}}}{(3^3)^{\frac{4}{3}}} = \frac{1}{3^{4}} = \frac{1}{81} \]

### Step 4: Rewrite the division as multiplication
Now we rewrite the division as multiplication by the reciprocal:
\[ \frac{1}{36} \div \frac{1}{81} = \frac{1}{36} \times \frac{81}{1} = \frac{81}{36} \]

### Step 5: Simplify the fraction
Now we simplify \(\frac{81}{36}\):
- Both numbers can be divided by \(9\):
\[ \frac{81 \div 9}{36 \div 9} = \frac{9}{4} \]

### Final Answer
Thus, the final answer is:
\[ \frac{9}{4} \]

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