The value of `(16)^((5)/(2)) + (16)^((-3)/(2))` is equal to

Options

A. `(66738)/(5)`
B. `(66635)/(46)`
C. `(65537)/(64)`
D. `(63537)/(64)`

Answer

To solve the expression \( (16)^{\frac{5}{2}} + (16)^{-\frac{3}{2}} \), we will break it down into manageable steps.

### Step 1: Simplify \( (16)^{\frac{5}{2}} \)

We can rewrite \( 16 \) as \( 4^2 \). Therefore, we have:

\[ (16)^{\frac{5}{2}} = (4^2)^{\frac{5}{2}} \]

Using the power of a power property \( (a^m)^n = a^{m \cdot n} \):

\[ (4^2)^{\frac{5}{2}} = 4^{2 \cdot \frac{5}{2}} = 4^5 \]

Now, calculating \( 4^5 \):

\[ 4^5 = 1024 \]

### Step 2: Simplify \( (16)^{-\frac{3}{2}} \)

Again, rewriting \( 16 \) as \( 4^2 \):

\[ (16)^{-\frac{3}{2}} = (4^2)^{-\frac{3}{2}} \]

Using the power of a power property:

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

Calculating \( 4^{-3} \):

\[ 4^{-3} = \frac{1}{4^3} = \frac{1}{64} \]

### Step 3: Combine the results

Now, we add the two results together:

\[ (16)^{\frac{5}{2}} + (16)^{-\frac{3}{2}} = 1024 + \frac{1}{64} \]

To add these, we need a common denominator. The common denominator between \( 1024 \) (which can be expressed as \( \frac{1024 \times 64}{64} \)) and \( \frac{1}{64} \) is \( 64 \):

\[ 1024 = \frac{1024 \times 64}{64} = \frac{65536}{64} \]

Now, adding them together:

\[ \frac{65536}{64} + \frac{1}{64} = \frac{65536 + 1}{64} = \frac{65537}{64} \]

### Final Answer

Thus, the value of \( (16)^{\frac{5}{2}} + (16)^{-\frac{3}{2}} \) is:

\[ \frac{65537}{64} \]

---