Perform the following by the indicated operations. Express the result in the form x+iy, where x,y are real number `i=sqrt(-1)`:
Simply:
(i) `(sqrt(5)+7i)(sqrt(5)-7i)^(3)`
(ii) `(1+i)(2+3i)(3+4i)(4+5i)`
(iii) `(1+i^(2))+i^(4)+i^(6)`.

Answer

Let's solve the given problems step by step.

### (i) Simplifying `(sqrt(5) + 7i)(sqrt(5) - 7i)^(3)`

1. **Calculate `(sqrt(5) - 7i)^(3)`**:
- First, find `(sqrt(5) - 7i)`.
- Use the binomial theorem or expand directly:
\[ (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 \]
where \( a = \sqrt{5} \) and \( b = 7i \).
- Calculate \( a^3 = (\sqrt{5})^3 = 5\sqrt{5} \).
- Calculate \( 3a^2b = 3(\sqrt{5})^2(7i) = 3(5)(7i) = 105i \).
- Calculate \( 3ab^2 = 3(\sqrt{5})(7i)^2 = 3(\sqrt{5})(-49) = -147\sqrt{5} \).
- Calculate \( b^3 = (7i)^3 = 343i^3 = -343i \).
- Combine these results:
\[ (sqrt{5} - 7i)^3 = 5\sqrt{5} - 105i - 147\sqrt{5} + 343i = (5\sqrt{5} - 147\sqrt{5}) + (-105i + 343i) \]
\[ = -142\sqrt{5} + 238i \]

2. **Now multiply `(sqrt(5) + 7i)` by the result**:
\[ (sqrt{5} + 7i)(-142\sqrt{5} + 238i) \]
- Use the distributive property:
\[ = sqrt{5}(-142\sqrt{5}) + sqrt{5}(238i) + 7i(-142\sqrt{5}) + 7i(238i) \]
- Calculate each term:
\[ = -142 \cdot 5 + 238\sqrt{5}i - 994\sqrt{5}i + 1666(-1) \]
\[ = -710 + 238\sqrt{5}i - 994\sqrt{5}i - 1666 \]
\[ = -2376 - 756\sqrt{5}i \]

3. **Final result**:
\[ = -2376 - 756\sqrt{5}i \]