Completing the Operation: (-a - 6b + 4)^3ab
In mathematics, we often encounter complex algebraic expressions that require careful step-by-step work to simplify and evaluate. One such expression is (-a - 6b + 4)^3ab. Let's break down the process of completing this operation.
1. **Understand the Expression**:
- The expression consists of two parts: (-a - 6b + 4) and 3ab.
- The first part, (-a - 6b + 4), is a polynomial expression with three terms: -a, -6b, and 4.
- The second part, 3ab, is a product of three factors: 3, a, and b.
- The operation to be performed is raising the first part to the power of the second part, which is ((-a - 6b + 4)^3ab).
2. **Simplify the First Part**:
- To simplify the first part, we need to apply the rules of exponents.
- The exponent 3ab means that the expression (-a - 6b + 4) should be raised to the power of 3ab.
- We can rewrite the expression as ((-a - 6b + 4)^3)^ab.
- Now, we can focus on simplifying the expression (-a - 6b + 4)^3.
- Using the rules of exponents, we can expand the expression: (-a - 6b + 4)^3 = (-a - 6b + 4) × (-a - 6b + 4) × (-a - 6b + 4).
3. **Multiply the Terms**:
- Multiplying the three factors, we get:
(-a - 6b + 4) × (-a - 6b + 4) × (-a - 6b + 4) = a^3 + 18a^2b - 72ab + 216b^2 - 12a^2 - 216ab + 288a - 1728b + 4096.
4. **Raise the Simplified Expression to the Power of ab**:
- Now that we have the simplified expression (-a - 6b + 4)^3, we need to raise it to the power of ab.
- Using the rules of exponents, we can write: (a^3 + 18a^2b - 72ab + 216b^2 - 12a^2 - 216ab + 288a - 1728b + 4096)^ab.
The final result of the operation (-a - 6b + 4)^3ab is the expression obtained by raising the simplified polynomial (a^3 + 18a^2b - 72ab + 216b^2 - 12a^2 - 216ab + 288a - 1728b + 4096) to the power of ab.
Remember, the key to successfully completing this operation is to understand the expression, simplify the first part, and then apply the rules of exponents to raise the simplified expression to the power of the second part.
