Simplify:
`[1/2^(2)-1/4^(3)]^-1xx2^(-3)`

Answer

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

### Step 1: Simplify the terms inside the brackets
First, we need to evaluate \(\frac{1}{2^2}\) and \(\frac{1}{4^3}\).

- \(2^2 = 4\), so \(\frac{1}{2^2} = \frac{1}{4}\).
- \(4^3 = 64\), so \(\frac{1}{4^3} = \frac{1}{64}\).

Now we can rewrite the expression:
\[ \left[\frac{1}{4} - \frac{1}{64}\right]^{-1} \times 2^{-3} \]

### Step 2: Find a common denominator and subtract
The common denominator for \(\frac{1}{4}\) and \(\frac{1}{64}\) is \(64\).

- Convert \(\frac{1}{4}\) to have a denominator of \(64\):
\[ \frac{1}{4} = \frac{16}{64} \]

Now, we can perform the subtraction:
\[ \frac{16}{64} - \frac{1}{64} = \frac{16 - 1}{64} = \frac{15}{64} \]

### Step 3: Substitute back into the expression
Now we substitute back into the expression:
\[ \left[\frac{15}{64}\right]^{-1} \times 2^{-3} \]

### Step 4: Apply the negative exponent
Using the property of exponents \(x^{-m} = \frac{1}{x^m}\), we have:
\[ \left[\frac{15}{64}\right]^{-1} = \frac{64}{15} \]

Now, the expression becomes:
\[ \frac{64}{15} \times 2^{-3} \]

### Step 5: Simplify \(2^{-3}\)
Calculating \(2^{-3}\):
\[ 2^{-3} = \frac{1}{2^3} = \frac{1}{8} \]

### Step 6: Multiply the fractions
Now we multiply:
\[ \frac{64}{15} \times \frac{1}{8} = \frac{64 \times 1}{15 \times 8} = \frac{64}{120} \]

### Step 7: Simplify the fraction
Now we can simplify \(\frac{64}{120}\):
- The greatest common divisor (GCD) of \(64\) and \(120\) is \(8\).
\[ \frac{64 \div 8}{120 \div 8} = \frac{8}{15} \]

### Final Answer
Thus, the simplified expression is:
\[ \frac{8}{15} \]

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