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The problem asks us to find the value of \( ab^3 \) given that:

- \( a = 5x \)
- \( b = -i \), where \( i \) is the imaginary unit, \( i = \sqrt{-1} \).

### Step 1: Calculate \( b^3 \)

We know that \( b = -i \). Let's calculate \( b^3 \):

\[
b^3 = (-i)^3 = -i^3
\]

Recall that \( i^3 = i \times i \times i = -i \), since \( i^2 = -1 \). Therefore:

\[
b^3 = -(-i) = i
\]

### Step 2: Multiply \( ab^3 \)

Now, using \( a = 5x \) and \( b^3 = i \), we can find \( ab^3 \):

\[
ab^3 = (5x) \times i = 5ix
\]

### Final Answer:

The value of \( ab^3 \) is \( 5ix \).

---

Would you like further details or have any questions?

Here are 5 related questions:
1. What is the general property of powers of \( i \) (i.e., \( i^1, i^2, i^3, i^4 \))?
2. What would happen if \( a \) had been a complex number instead of \( 5x \)?
3. How does the imaginary unit \( i \) interact with real numbers in multiplication?
4. How would \( ab^3 \) change if \( b = 2i \) instead of \( -i \)?
5. Can we extend this approach to higher powers of \( i \), such as \( i^5 \) or \( i^6 \)?

**Tip:** Powers of the imaginary unit follow a cyclical pattern every four terms: \( i^1 = i \), \( i^2 = -1 \), \( i^3 = -i \), and \( i^4 = 1 \).

If a = 5x and b = -i, then find the value of ab^3 in fully simplified form.

This math problem involves finding the value of ab^3 given that a = 5x and b = -i. Learn how to simplify powers of i and multiply real numbers with imaginary units. Suitable for high school students in Grades 9-12.

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