Find the term independent of x in the expansion of
`((2x^(2))/(5)+(15)/(4x))^(9)`

Answer

To find the term independent of \( x \) in the expansion of

\[ \left( \frac{2x^2}{5} + \frac{15}{4x} \right)^9, \]

we will follow these steps:

### Step 1: Identify the General Term
The general term \( T_r \) in the binomial expansion of \( (a + b)^n \) is given by:

\[ T_r = \binom{n}{r} a^{n-r} b^r \]

In our case, \( a = \frac{2x^2}{5} \), \( b = \frac{15}{4x} \), and \( n = 9 \). Therefore, the general term is:

\[ T_r = \binom{9}{r} \left( \frac{2x^2}{5} \right)^{9-r} \left( \frac{15}{4x} \right)^r \]

### Step 2: Simplify the General Term
Now we simplify \( T_r \):

\[ T_r = \binom{9}{r} \left( \frac{2^{9-r} x^{2(9-r)}}{5^{9-r}} \right) \left( \frac{15^r}{4^r x^r} \right) \]

This can be rewritten as:

\[ T_r = \binom{9}{r} \frac{2^{9-r} \cdot 15^r}{5^{9-r} \cdot 4^r} x^{2(9-r) - r} \]

### Step 3: Find the Power of \( x \)
The exponent of \( x \) in \( T_r \) is:

\[ 2(9-r) - r = 18 - 2r - r = 18 - 3r \]

### Step 4: Set the Exponent of \( x \) to Zero
To find the term independent of \( x \), we set the exponent equal to zero:

\[ 18 - 3r = 0 \]

Solving for \( r \):

\[ 3r = 18 \implies r = 6 \]

### Step 5: Substitute \( r \) Back into the General Term
Now we substitute \( r = 6 \) back into the general term \( T_r \):

\[ T_6 = \binom{9}{6} \left( \frac{2}{5} \right)^{9-6} \left( \frac{15}{4} \right)^6 \]

This simplifies to:

\[ T_6 = \binom{9}{6} \left( \frac{2}{5} \right)^3 \left( \frac{15}{4} \right)^6 \]

### Step 6: Calculate \( T_6 \)
Now we calculate \( T_6 \):

1. Calculate \( \binom{9}{6} = \binom{9}{3} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84 \).
2. Calculate \( \left( \frac{2}{5} \right)^3 = \frac{8}{125} \).
3. Calculate \( \left( \frac{15}{4} \right)^6 = \frac{15^6}{4^6} = \frac{11390625}{4096} \).

Putting it all together:

\[ T_6 = 84 \cdot \frac{8}{125} \cdot \frac{11390625}{4096} \]

### Step 7: Final Calculation
Now we simplify:

\[ T_6 = \frac{84 \cdot 8 \cdot 11390625}{125 \cdot 4096} \]

Calculating this gives the term independent of \( x \).