Expand using binomial theorem `(4x - 5y)^5 ` = ………..

Answer

To expand the expression \((4x - 5y)^5\) using the Binomial Theorem, we follow these steps:

### Step 1: Identify the terms
In the expression \((4x - 5y)^5\), we can identify:
- \(a = 4x\)
- \(b = -5y\)
- \(n = 5\)

### Step 2: Write the Binomial Theorem formula
The Binomial Theorem states that:
\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]
For our case, we have:
\[ (4x - 5y)^5 = \sum_{k=0}^{5} \binom{5}{k} (4x)^{5-k} (-5y)^k \]

### Step 3: Calculate each term in the expansion
We will calculate each term for \(k = 0\) to \(k = 5\):

1. **For \(k = 0\)**:
\[ \binom{5}{0} (4x)^5 (-5y)^0 = 1 \cdot (4x)^5 \cdot 1 = 1024x^5 \]

2. **For \(k = 1\)**:
\[ \binom{5}{1} (4x)^4 (-5y)^1 = 5 \cdot (256x^4) \cdot (-5y) = -12800x^4y \]

3. **For \(k = 2\)**:
\[ \binom{5}{2} (4x)^3 (-5y)^2 = 10 \cdot (64x^3) \cdot 25y^2 = 16000x^3y^2 \]

4. **For \(k = 3\)**:
\[ \binom{5}{3} (4x)^2 (-5y)^3 = 10 \cdot (16x^2) \cdot (-125y^3) = -20000x^2y^3 \]

5. **For \(k = 4\)**:
\[ \binom{5}{4} (4x)^1 (-5y)^4 = 5 \cdot (4x) \cdot 625y^4 = 12500xy^4 \]

6. **For \(k = 5\)**:
\[ \binom{5}{5} (4x)^0 (-5y)^5 = 1 \cdot 1 \cdot (-3125y^5) = -3125y^5 \]

### Step 4: Combine all terms
Now, we combine all the calculated terms:
\[ (4x - 5y)^5 = 1024x^5 - 12800x^4y + 16000x^3y^2 - 20000x^2y^3 + 12500xy^4 - 3125y^5 \]

### Final Answer:
\[ (4x - 5y)^5 = 1024x^5 - 12800x^4y + 16000x^3y^2 - 20000x^2y^3 + 12500xy^4 - 3125y^5 \]