Simplify the complex number and write it in standard form.
−8i3 + i2

To simplify the expression \( -8i^3 + i^2 \), we'll use the properties of the imaginary unit \( i \), where:

\[
i^2 = -1 \quad \text{and} \quad i^3 = -i
\]

### Step-by-step simplification:

1. Simplify \( i^2 \):
   \[
   i^2 = -1
   \]

2. Simplify \( i^3 \):
   \[
   i^3 = -i
   \]

3. Substitute these values back into the expression:
   \[
   -8i^3 + i^2 = -8(-i) + (-1)
   \]

4. Simplify:
   \[
   -8(-i) + (-1) = 8i - 1
   \]

### Final Answer:
The simplified expression in standard form is:
\[
8i - 1
\]

The standard form of a complex number is \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. So the final answer is:

\[
-1 + 8i
\]

Would you like more details on any part of this solution? 

Here are five related questions to explore:

1. What are the higher powers of \( i \) (e.g., \( i^4, i^5 \))?
2. How do you multiply two complex numbers?
3. How do you find the magnitude of a complex number?
4. What is the conjugate of a complex number?
5. How do you add or subtract complex numbers?

**Tip:** Memorizing the cycle of powers of \( i \) helps simplify complex number expressions easily!

Simplify the complex number and write it in standard form: −8i^3 + i^2

Learn how to simplify the complex number −8i^3 + i^2 and express it in standard form. This solution explains the properties of the imaginary unit, including i^2 = -1 and i^3 = -i, and is suitable for students in Grades 10-12.

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