(2x%5E-3y)%5E-1.jpeg "(12x^3-16x^2y+3xy^2+9y^2)(2x^-3y)^-1")
Simplifying the Expression: (12x^3 - 16x^2y + 3xy^2 + 9y^2)(2x^-3y)^-1
This problem involves simplifying an expression with both polynomial and exponential terms. Let's break down the steps:
1. Understanding the Problem:
The expression is a product of two factors:
- A polynomial: 12x^3 - 16x^2y + 3xy^2 + 9y^2
- A term with negative exponents: (2x^-3y)^-1
2. Simplifying the Exponent:
First, we need to simplify the term with the negative exponents:
- (2x^-3y)^-1 = 1 / (2x^-3y)
Remember, a term raised to a negative power is the same as its reciprocal raised to the positive version of that power.
3. Applying the Negative Exponent:
- 1 / (2x^-3y) = 1 / (2 * (1/x^3) * y) = x^3 / (2y)
Here, we use the rule that x^-n = 1/x^n.
4. Multiplying the Polynomial by the Simplified Term:
Now we multiply the polynomial by the simplified term:
(12x^3 - 16x^2y + 3xy^2 + 9y^2) * (x^3 / (2y))
We can distribute the x^3 / (2y) term to each term in the polynomial:
= (12x^3 * x^3) / (2y) - (16x^2y * x^3) / (2y) + (3xy^2 * x^3) / (2y) + (9y^2 * x^3) / (2y)
5. Simplifying the Expression:
Finally, we simplify the expression by combining like terms and applying the rules of exponents:
= 6x^6 / y - 8x^5 - (3/2)x^4y + (9/2)x^3y
Therefore, the simplified form of (12x^3 - 16x^2y + 3xy^2 + 9y^2)(2x^-3y)^-1 is 6x^6 / y - 8x^5 - (3/2)x^4y + (9/2)x^3y.