`(5z^(6)+1/(z^(6)))^(2)=` ____

Options

A. `5z^(8)+10z^(6)+1/(z^(8))`
B. `25z^(12)+10z^(6)+1/(z^(12))`
C. `25z^(12)+10+1/(z^(12))`
D. `5z^(8)+10+1/(z^(8))`

Answer

To solve the expression \((5z^{6} + \frac{1}{z^{6}})^{2}\), we can use the algebraic identity for the square of a binomial, which states:

\[ (A + B)^{2} = A^{2} + B^{2} + 2AB \]

### Step-by-Step Solution:

1. **Identify A and B**:
- Let \(A = 5z^{6}\) and \(B = \frac{1}{z^{6}}\).

2. **Apply the identity**:
- Using the identity, we can expand \((A + B)^{2}\):
\[ (5z^{6} + \frac{1}{z^{6}})^{2} = (5z^{6})^{2} + \left(\frac{1}{z^{6}}\right)^{2} + 2(5z^{6})(\frac{1}{z^{6}}) \]

3. **Calculate \(A^{2}\)**:
- \((5z^{6})^{2} = 25z^{12}\)

4. **Calculate \(B^{2}\)**:
- \(\left(\frac{1}{z^{6}}\right)^{2} = \frac{1}{z^{12}}\)

5. **Calculate \(2AB\)**:
- \(2(5z^{6})(\frac{1}{z^{6}}) = 2 \cdot 5 = 10\)

6. **Combine all parts**:
- Now, we can combine all the parts together:
\[ (5z^{6} + \frac{1}{z^{6}})^{2} = 25z^{12} + \frac{1}{z^{12}} + 10 \]

### Final Answer:
\[ (5z^{6} + \frac{1}{z^{6}})^{2} = 25z^{12} + \frac{1}{z^{12}} + 10 \]

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