Expand `(2x^(2) + 4y^(2))^(6)` using pascals triangle.

Answer

To expand the expression \((2x^2 + 4y^2)^6\) using Pascal's Triangle, we will follow these steps:

### Step 1: Identify the coefficients from Pascal's Triangle
For \(n = 6\), the coefficients from Pascal's Triangle are:
\[ 1, 6, 15, 20, 15, 6, 1 \]

### Step 2: Write the general term of the binomial expansion
The general term in the binomial expansion of \((a + b)^n\) is given by:
\[ T_k = C(n, k) \cdot a^{n-k} \cdot b^k \]
where \(C(n, k)\) is the binomial coefficient.

In our case, \(a = 2x^2\), \(b = 4y^2\), and \(n = 6\). Thus, the general term becomes:
\[ T_k = C(6, k) \cdot (2x^2)^{6-k} \cdot (4y^2)^k \]

### Step 3: Calculate each term for \(k = 0\) to \(k = 6\)
Now we will calculate each term for \(k\) from 0 to 6:

1. **For \(k = 0\)**:
\[ T_0 = C(6, 0) \cdot (2x^2)^{6} \cdot (4y^2)^{0} = 1 \cdot (2x^2)^{6} = 64x^{12} \]

2. **For \(k = 1\)**:
\[ T_1 = C(6, 1) \cdot (2x^2)^{5} \cdot (4y^2)^{1} = 6 \cdot (2x^2)^{5} \cdot (4y^2) = 6 \cdot 32x^{10} \cdot 4y^2 = 768x^{10}y^2 \]

3. **For \(k = 2\)**:
\[ T_2 = C(6, 2) \cdot (2x^2)^{4} \cdot (4y^2)^{2} = 15 \cdot (2x^2)^{4} \cdot (16y^4) = 15 \cdot 16x^8 \cdot 16y^4 = 3840x^8y^4 \]

4. **For \(k = 3\)**:
\[ T_3 = C(6, 3) \cdot (2x^2)^{3} \cdot (4y^2)^{3} = 20 \cdot (8x^6) \cdot (64y^6) = 1280x^6y^6 \]

5. **For \(k = 4\)**:
\[ T_4 = C(6, 4) \cdot (2x^2)^{2} \cdot (4y^2)^{4} = 15 \cdot (4x^4) \cdot (256y^8) = 9600x^4y^8 \]

6. **For \(k = 5\)**:
\[ T_5 = C(6, 5) \cdot (2x^2)^{1} \cdot (4y^2)^{5} = 6 \cdot (2x^2) \cdot (1024y^{10}) = 12288x^2y^{10} \]

7. **For \(k = 6\)**:
\[ T_6 = C(6, 6) \cdot (2x^2)^{0} \cdot (4y^2)^{6} = 1 \cdot (4096y^{12}) = 4096y^{12} \]

### Step 4: Combine all the terms
Now, we combine all the terms to get the final expanded form:
\[ (2x^2 + 4y^2)^6 = 64x^{12} + 768x^{10}y^2 + 3840x^8y^4 + 1280x^6y^6 + 9600x^4y^8 + 12288x^2y^{10} + 4096y^{12} \]

### Final Answer:
\[ (2x^2 + 4y^2)^6 = 64x^{12} + 768x^{10}y^2 + 3840x^8y^4 + 1280x^6y^6 + 9600x^4y^8 + 12288x^2y^{10} + 4096y^{12} \]