Write the following in the form x+iy:
(i) `(2i)^(3)`
(ii) `i^(-35)`
(iii) `(-i)(2i)(-(1)/(8)i)^(3)`.

Answer

Let's solve the given problems step by step.

### (i) \( (2i)^3 \)

1. **Calculate \( (2i)^3 \)**:
\[ (2i)^3 = 2^3 \cdot i^3 = 8 \cdot i^3 \]

2. **Substitute \( i^3 \)**:
Since \( i^3 = i^2 \cdot i = (-1) \cdot i = -i \):
\[ 8 \cdot i^3 = 8 \cdot (-i) = -8i \]

3. **Write in the form \( x + iy \)**:
\[ -8i = 0 - 8i \]
Thus, the answer is \( 0 - 8i \).

### (ii) \( i^{-35} \)

1. **Rewrite \( i^{-35} \)**:
\[ i^{-35} = \frac{1}{i^{35}} \]

2. **Express \( i^{35} \)**:
Since \( i^4 = 1 \), we can find \( i^{35} \) by reducing the exponent modulo 4:
\[ 35 \mod 4 = 3 \quad \Rightarrow \quad i^{35} = i^3 \]

3. **Substitute \( i^3 \)**:
\[ i^{35} = -i \]
So,
\[ i^{-35} = \frac{1}{-i} \]

4. **Multiply numerator and denominator by \( i \)**:
\[ i^{-35} = \frac{i}{-i^2} = \frac{i}{1} = i \]

5. **Write in the form \( x + iy \)**:
\[ i = 0 + 1i \]
Thus, the answer is \( 0 + 1i \).

### (iii) \( (-i)(2i)\left(-\frac{1}{8}i\right)^3 \)

1. **Calculate \( \left(-\frac{1}{8}i\right)^3 \)**:
\[ \left(-\frac{1}{8}i\right)^3 = -\frac{1}{8^3}i^3 = -\frac{1}{512}(-i) = \frac{1}{512}i \]

2. **Combine the terms**:
\[ (-i)(2i)\left(\frac{1}{512}i\right) \]

3. **Calculate \( (-i)(2i) \)**:
\[ (-i)(2i) = -2i^2 = -2(-1) = 2 \]

4. **Now multiply by \( \frac{1}{512}i \)**:
\[ 2 \cdot \frac{1}{512}i = \frac{2}{512}i = \frac{1}{256}i \]

5. **Write in the form \( x + iy \)**:
\[ \frac{1}{256}i = 0 + \frac{1}{256}i \]
Thus, the answer is \( 0 + \frac{1}{256}i \).

### Final Answers:
1. \( (2i)^3 = 0 - 8i \)
2. \( i^{-35} = 0 + 1i \)
3. \( (-i)(2i)\left(-\frac{1}{8}i\right)^3 = 0 + \frac{1}{256}i \)