Find the value of `4^(3)+(1)/(4^3)`

Options

A. `(4043)/(64)`
B. `(2049)/(64)`
C. `(4097)/(64)`
D. `(2093)/(64)`

Answer

To find the value of \(4^3 + \frac{1}{4^3}\), we can use the algebraic identity for the sum of cubes. Let's break it down step by step.

### Step-by-Step Solution:

1. **Calculate \(4^3\)**:
\[ 4^3 = 4 \times 4 \times 4 = 64 \]

2. **Calculate \(\frac{1}{4^3}\)**:
\[ \frac{1}{4^3} = \frac{1}{64} \]

3. **Combine the two results**:
\[ 4^3 + \frac{1}{4^3} = 64 + \frac{1}{64} \]

4. **Convert 64 to a fraction with a common denominator**:
\[ 64 = \frac{64 \times 64}{64} = \frac{4096}{64} \]

5. **Add the two fractions**:
\[ \frac{4096}{64} + \frac{1}{64} = \frac{4096 + 1}{64} = \frac{4097}{64} \]

### Final Answer:
\[ 4^3 + \frac{1}{4^3} = \frac{4097}{64} \]