Simplify : `(((-5)/(2))^(7)div((-5)/(2))^3)/(((4)/(7))^(4)div((4)/(7))^3)`

Answer

To simplify the expression \(\frac{(-\frac{5}{2})^7 \div (-\frac{5}{2})^3}{(\frac{4}{7})^4 \div (\frac{4}{7})^3}\), we can follow these steps:

### Step 1: Apply the property of exponents
Using the property of exponents that states \(\frac{a^m}{a^n} = a^{m-n}\), we can simplify both the numerator and the denominator.

**Numerator:**
\[ (-\frac{5}{2})^7 \div (-\frac{5}{2})^3 = (-\frac{5}{2})^{7-3} = (-\frac{5}{2})^4 \]

**Denominator:**
\[ (\frac{4}{7})^4 \div (\frac{4}{7})^3 = (\frac{4}{7})^{4-3} = (\frac{4}{7})^1 = \frac{4}{7} \]

### Step 2: Rewrite the expression
Now, we can rewrite the original expression as:
\[ \frac{(-\frac{5}{2})^4}{\frac{4}{7}} \]

### Step 3: Simplify the fraction
To simplify the fraction, we can multiply by the reciprocal of the denominator:
\[ (-\frac{5}{2})^4 \times \frac{7}{4} \]

### Step 4: Calculate \((-5/2)^4\)
Calculating \((-5/2)^4\):
\[ (-5)^4 = 625 \quad \text{and} \quad (2)^4 = 16 \]
Thus,
\[ (-\frac{5}{2})^4 = \frac{625}{16} \]

### Step 5: Combine the results
Now we can combine the results:
\[ \frac{625}{16} \times \frac{7}{4} = \frac{625 \times 7}{16 \times 4} \]

### Step 6: Calculate the final result
Calculating the numerator and denominator:
\[ 625 \times 7 = 4375 \quad \text{and} \quad 16 \times 4 = 64 \]
Thus, the final result is:
\[ \frac{4375}{64} \]

### Final Answer
The simplified expression is:
\[ \frac{4375}{64} \]