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

Options

A. (a) 19
B. (b) `frac(1)(19)`
C. (c) -19
D. (d) `frac(-1)(19)`

Answer

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

### Step 1: Rewrite the expression using positive exponents
Using the property of exponents that states \(a^{-n} = \frac{1}{a^n}\), we can rewrite the expression:

\[ \left(\frac{1}{3}\right)^{-3} = 3^3 \quad \text{and} \quad \left(\frac{1}{2}\right)^{-3} = 2^3 \]

So, the expression becomes:

\[ 3^3 - 2^3 \]

### Step 2: Calculate the values of \(3^3\) and \(2^3\)
Now, we calculate \(3^3\) and \(2^3\):

\[ 3^3 = 3 \times 3 \times 3 = 27 \]

\[ 2^3 = 2 \times 2 \times 2 = 8 \]

### Step 3: Substitute the values back into the expression
Now substitute these values back into the expression:

\[ 27 - 8 \]

### Step 4: Perform the subtraction
Now, we perform the subtraction:

\[ 27 - 8 = 19 \]

### Final Answer
Thus, the final answer is:

\[ \boxed{19} \]

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