Find the value of `{ ((1)/(3))^(-2) - ((1)/(2))^(-3) } div ((1)/(4))^(-2)`

Options

A. `16//3`
B. `4^(-2)`
C. `4^2`
D. `4^4`

Answer

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

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

\[ \left( \frac{1}{3} \right)^{-2} = 3^2 \]
\[ \left( \frac{1}{2} \right)^{-3} = 2^3 \]
\[ \left( \frac{1}{4} \right)^{-2} = 4^2 \]

### Step 2: Substitute the rewritten values
Now we substitute these values back into the expression:

\[ \frac{3^2 - 2^3}{4^2} \]

### Step 3: Calculate the powers
Next, we calculate the powers:

\[ 3^2 = 9 \]
\[ 2^3 = 8 \]
\[ 4^2 = 16 \]

### Step 4: Substitute the calculated values
Now we substitute these calculated values into the expression:

\[ \frac{9 - 8}{16} \]

### Step 5: Simplify the numerator
Now we simplify the numerator:

\[ 9 - 8 = 1 \]

### Step 6: Final simplification
Now we have:

\[ \frac{1}{16} \]

### Conclusion
Thus, the final value of the expression is:

\[ \frac{1}{16} \]

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