Simplify `(1024)^(-3//5)`.

Answer

To simplify \( (1024)^{-\frac{3}{5}} \), we can follow these steps:

### Step 1: Change the Negative Exponent
The first step is to change the negative exponent to a positive exponent. According to the rule of exponents, \( a^{-n} = \frac{1}{a^n} \).

\[ (1024)^{-\frac{3}{5}} = \frac{1}{(1024)^{\frac{3}{5}}} \]

### Step 2: Express 1024 as a Power of 4
Next, we need to express 1024 in terms of a base that is easier to work with. We know that \( 1024 = 4^5 \).

\[ (1024)^{\frac{3}{5}} = (4^5)^{\frac{3}{5}} \]

### Step 3: Apply the Power of a Power Rule
Using the power of a power rule, \( (a^m)^n = a^{m \cdot n} \), we can simplify further:

\[ (4^5)^{\frac{3}{5}} = 4^{5 \cdot \frac{3}{5}} = 4^3 \]

### Step 4: Calculate \( 4^3 \)
Now we can calculate \( 4^3 \):

\[ 4^3 = 4 \times 4 \times 4 = 64 \]

### Step 5: Substitute Back
Now substitute back into the expression we had from Step 1:

\[ (1024)^{-\frac{3}{5}} = \frac{1}{(1024)^{\frac{3}{5}}} = \frac{1}{4^3} = \frac{1}{64} \]

### Final Answer
Thus, the simplified form of \( (1024)^{-\frac{3}{5}} \) is:

\[ \frac{1}{64} \]

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