The sum of the binomial coefficients in the expansion of `(x^(-3/4) + ax^(5/4))^n` lies between 200 and 400 and the term independent of x equals 448. The value of a is :-

Answer

To find the value of \( a \) in the expression \( (x^{-\frac{3}{4}} + ax^{\frac{5}{4}})^n \) given that the sum of the binomial coefficients lies between 200 and 400 and the term independent of \( x \) equals 448, we can follow these steps:

### Step 1: Understand the Binomial Expansion
The binomial expansion of \( (x^{-\frac{3}{4}} + ax^{\frac{5}{4}})^n \) can be expressed as:
\[ \sum_{r=0}^{n} \binom{n}{r} (x^{-\frac{3}{4}})^{n-r} (ax^{\frac{5}{4}})^{r} \]
This simplifies to:
\[ \sum_{r=0}^{n} \binom{n}{r} a^r x^{-\frac{3(n-r)}{4} + \frac{5r}{4}} \]

### Step 2: Find the Term Independent of \( x \)
The term independent of \( x \) occurs when the exponent of \( x \) is zero:
\[ -\frac{3(n-r)}{4} + \frac{5r}{4} = 0 \]
Solving for \( r \):
\[ -\frac{3n}{4} + \frac{3r}{4} + \frac{5r}{4} = 0 \]
\[ -\frac{3n}{4} + \frac{8r}{4} = 0 \implies 8r = 3n \implies r = \frac{3n}{8} \]

### Step 3: Ensure \( r \) is an Integer
For \( r \) to be an integer, \( n \) must be a multiple of 8. Let \( n = 8k \) for some integer \( k \).

### Step 4: Calculate the Independent Term
The term independent of \( x \) is given by:
\[ T_r = \binom{n}{r} a^r x^{0} = \binom{n}{\frac{3n}{8}} a^{\frac{3n}{8}} \]
Substituting \( n = 8k \):
\[ T_r = \binom{8k}{3k} a^{3k} \]
Given that this term equals 448, we have:
\[ \binom{8k}{3k} a^{3k} = 448 \]

### Step 5: Calculate the Sum of Binomial Coefficients
The sum of the binomial coefficients is:
\[ \sum_{r=0}^{n} \binom{n}{r} = 2^n \]
Thus:
\[ 200 < 2^n < 400 \]
Calculating the powers of 2:
- \( 2^7 = 128 \) (too low)
- \( 2^8 = 256 \) (valid)
- \( 2^9 = 512 \) (too high)

So, \( n = 8 \).

### Step 6: Substitute \( n \) into the Independent Term Equation
Now substituting \( n = 8 \):
\[ \binom{8}{3} a^3 = 448 \]
Calculating \( \binom{8}{3} \):
\[ \binom{8}{3} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56 \]
Thus, we have:
\[ 56 a^3 = 448 \]
Dividing both sides by 56:
\[ a^3 = \frac{448}{56} = 8 \]
Taking the cube root:
\[ a = 2 \]

### Final Answer
The value of \( a \) is \( \boxed{2} \).