If a, equals, minus, 2, x, plus, ia=−2x+i and b, equals, 4, ib=4i, then find the value of the a, b, cubedab 
3
  in fully simplified form.

Given that:

\[
a = -2x + i
\]
\[
b = 4i
\]

We are tasked with finding the value of \( ab^3 \), where \( ab^3 \) means the product of \( a \) and the cube of \( b \).

### Step 1: Cube \( b \)
Since \( b = 4i \), let's first calculate \( b^3 \):

\[
b^3 = (4i)^3 = 4^3 \cdot i^3 = 64 \cdot i^3
\]
Now recall that \( i^2 = -1 \) and \( i^3 = i^2 \cdot i = -i \), so:

\[
b^3 = 64 \cdot (-i) = -64i
\]

### Step 2: Multiply \( a \) by \( b^3 \)
Now, we multiply \( a = -2x + i \) by \( b^3 = -64i \):

\[
ab^3 = (-2x + i)(-64i)
\]

Distribute \( -64i \) across the terms in \( a \):

\[
ab^3 = (-2x)(-64i) + (i)(-64i)
\]

\[
ab^3 = 128xi + (-64i^2)
\]

Since \( i^2 = -1 \), we have:

\[
-64i^2 = -64(-1) = 64
\]

Thus, the expression simplifies to:

\[
ab^3 = 128xi + 64
\]

### Final Answer:
The fully simplified form of \( ab^3 \) is:

\[
ab^3 = 128xi + 64
\]

Would you like further clarification or details on any step? Here are five related questions:

1. What are the properties of imaginary numbers like \( i \)?
2. How do you expand complex number expressions?
3. Can you explain the process of multiplying complex numbers?
4. What happens when you cube a complex number?
5. How do powers of \( i \) cycle through values?

**Tip:** Always remember the cycle of powers of \( i \) (i.e., \( i^1 = i \), \( i^2 = -1 \), \( i^3 = -i \), \( i^4 = 1 \), and so on) when dealing with complex numbers.

If a = −2x + i and b = 4i, then find the value of ab^3 in fully simplified form.

Learn how to simplify the expression ab^3 where a = −2x + i and b = 4i. This problem covers key algebraic concepts and the use of powers of imaginary numbers. Suitable for advanced students in Grades 11-12.

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