`{(frac(1)(3))^(-3)-(frac(1)(2))^(-3)} div (frac(1)(4))^(-3)=` ?

Options

A. `frac(19)(64)`
B. `frac(64)(19)`
C. `frac(27)(16)`
D. none of these

Answer

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

### Step 1: Rewrite the expression
We start with the expression:
\[ \left(\left(\frac{1}{3}\right)^{-3} - \left(\frac{1}{2}\right)^{-3}\right) \div \left(\frac{1}{4}\right)^{-3} \]

### Step 2: Convert negative exponents to positive
Recall that \(a^{-n} = \frac{1}{a^n}\). Therefore, we can rewrite the terms with negative exponents:
\[ \left(\left(\frac{1}{3}\right)^{-3} = 3^3\right) \quad \text{and} \quad \left(\frac{1}{2}\right)^{-3} = 2^3 \quad \text{and} \quad \left(\frac{1}{4}\right)^{-3} = 4^3 \]

Thus, we can rewrite the expression as:
\[ \left(3^3 - 2^3\right) \div 4^3 \]

### Step 3: Calculate the powers
Now we calculate the powers:
\[ 3^3 = 27, \quad 2^3 = 8, \quad \text{and} \quad 4^3 = 64 \]

### Step 4: Substitute the values back into the expression
Substituting these values back, we have:
\[ \left(27 - 8\right) \div 64 \]

### Step 5: Perform the subtraction
Now, we perform the subtraction:
\[ 27 - 8 = 19 \]

### Step 6: Divide by \(64\)
Now we divide by \(64\):
\[ \frac{19}{64} \]

### Final Answer
Thus, the final answer is:
\[ \frac{19}{64} \]

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