The simplified form of `(16^(3//2)+16^(-3//2))` is :

Options

A. `0`
B. `(4097)/(64)`
C. `1`
D. `(16)/(4097)`

Answer

To simplify the expression \(16^{\frac{3}{2}} + 16^{-\frac{3}{2}}\), we can follow these steps:

### Step 1: Rewrite the powers of 16
We know that \(16\) can be expressed as \(4^2\). Therefore, we can rewrite \(16^{\frac{3}{2}}\) as:
\[ (4^2)^{\frac{3}{2}} = 4^{2 \cdot \frac{3}{2}} = 4^3 \]

### Step 2: Calculate \(4^3\)
Now, we calculate \(4^3\):
\[ 4^3 = 64 \]

### Step 3: Rewrite \(16^{-\frac{3}{2}}\)
Next, we simplify \(16^{-\frac{3}{2}}\):
\[ 16^{-\frac{3}{2}} = \frac{1}{16^{\frac{3}{2}}} = \frac{1}{4^3} = \frac{1}{64} \]

### Step 4: Combine the results
Now we can combine the results from Step 2 and Step 3:
\[ 16^{\frac{3}{2}} + 16^{-\frac{3}{2}} = 64 + \frac{1}{64} \]

### Step 5: Find a common denominator
To add these two terms, we need a common denominator:
\[ 64 = \frac{64 \times 64}{64} = \frac{4096}{64} \]
Thus,
\[ 64 + \frac{1}{64} = \frac{4096}{64} + \frac{1}{64} = \frac{4096 + 1}{64} = \frac{4097}{64} \]

### Final Answer
The simplified form of \(16^{\frac{3}{2}} + 16^{-\frac{3}{2}}\) is:
\[ \frac{4097}{64} \]

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