The value of `(8.(3.75)^3+1)/((7.5)^(2)-6.5)` is

Options

A. 2.75
B. `9/5`
C. 4.75
D. 8.5

Answer

To solve the expression \((8 \cdot (3.75)^3 + 1) / ((7.5)^2 - 6.5)\), we will follow these steps:

### Step 1: Calculate \( (3.75)^3 \)
First, we need to find the cube of \(3.75\):
\[ 3.75 = \frac{15}{4} \]
Now, calculate \((3.75)^3\):
\[ (3.75)^3 = \left(\frac{15}{4}\right)^3 = \frac{15^3}{4^3} = \frac{3375}{64} \]

### Step 2: Calculate \( 8 \cdot (3.75)^3 + 1 \)
Now, we substitute \((3.75)^3\) back into the expression:
\[ 8 \cdot (3.75)^3 + 1 = 8 \cdot \frac{3375}{64} + 1 \]
Calculating \(8 \cdot \frac{3375}{64}\):
\[ = \frac{8 \cdot 3375}{64} = \frac{27000}{64} \]
Now, we need to add \(1\) (which is \(\frac{64}{64}\)):
\[ \frac{27000}{64} + \frac{64}{64} = \frac{27000 + 64}{64} = \frac{27064}{64} \]

### Step 3: Calculate \( (7.5)^2 - 6.5 \)
Next, we calculate \((7.5)^2\):
\[ 7.5 = \frac{15}{2} \]
So,
\[ (7.5)^2 = \left(\frac{15}{2}\right)^2 = \frac{225}{4} \]
Now, subtract \(6.5\) (which is \(\frac{13}{2}\)):
\[ (7.5)^2 - 6.5 = \frac{225}{4} - \frac{13}{2} = \frac{225}{4} - \frac{26}{4} = \frac{225 - 26}{4} = \frac{199}{4} \]

### Step 4: Combine the results into the main expression
Now we can substitute these results back into the main expression:
\[ \frac{8 \cdot (3.75)^3 + 1}{(7.5)^2 - 6.5} = \frac{\frac{27064}{64}}{\frac{199}{4}} \]
To divide by a fraction, we multiply by its reciprocal:
\[ = \frac{27064}{64} \cdot \frac{4}{199} = \frac{27064 \cdot 4}{64 \cdot 199} \]
Now simplify:
\[ = \frac{27064}{16 \cdot 199} = \frac{27064}{3184} \]

### Step 5: Simplify the fraction
Now we simplify \(\frac{27064}{3184}\):
\[ \frac{27064 \div 3184}{3184 \div 3184} = \frac{8.5}{1} = 8.5 \]

### Final Answer
Thus, the value of the expression is:
\[ \boxed{8.5} \]