Simplify (12x^3-16x^2y+3xy^2+9y^2)(2x^-3y)^-1
The question is asking for the simplification of a mathematical expression that consists of two parts. The first part is a polynomial with four terms (12x^3-16x^2y+3xy^2+9y^2), and the second part is a monomial raised to the power of negative one (2x^-3y)^-1, which is mathematically equivalent to taking the reciprocal of the monomial. The task is to perform the necessary algebraic operations to combine these two expressions into a simpler form, which means distributing the polynomial by the reciprocal of the monomial and simplifying the result.
$\left(\right. 12 x^{3} - 16 x^{2} y + 3 x y^{2} + 9 y^{2} \left.\right) \left(\left(\right. 2 x^{- 3} y \left.\right)\right)^{- 1}$
Answer
Solution:
Step 1:
Apply the negative exponent rule which states that $a^{-n} = \frac{1}{a^n}$.
$(12x^3 - 16x^2y + 3xy^2 + 9y^2)(2x^{-3}y)^{-1}$
Step 2:
Combine the constants 2 and $x^{-3}$.
$(12x^3 - 16x^2y + 3xy^2 + 9y^2)\left(\frac{2}{x^3}y\right)^{-1}$
Step 3:
Merge the terms $\frac{2}{x^3}$ and y.
$(12x^3 - 16x^2y + 3xy^2 + 9y^2)\left(\frac{2y}{x^3}\right)^{-1}$
Step 4:
Invert the base to change the negative exponent to a positive one.
$(12x^3 - 16x^2y + 3xy^2 + 9y^2)\frac{x^3}{2y}$
Step 5:
Distribute $\frac{x^3}{2y}$ across the polynomial.
$\frac{(12x^3 - 16x^2y + 3xy^2 + 9y^2)x^3}{2y}$
Step 6:
Rearrange the factors in the expression.
$\frac{x^3(12x^3 - 16x^2y + 3xy^2 + 9y^2)}{2y}$
Knowledge Notes:
Negative Exponent Rule: For any non-zero number $a$ and any integer $n$, $a^{-n} = \frac{1}{a^n}$. This rule is used to simplify expressions with negative exponents by rewriting them as fractions with positive exponents.
Combining Like Terms: When simplifying algebraic expressions, terms with the same variables raised to the same powers can be combined by adding or subtracting their coefficients.
Distributive Property: This property states that $a(b + c) = ab + ac$. It is used to multiply a single term by each term in a polynomial.
Multiplication of Fractions: To multiply fractions, multiply the numerators together to get the new numerator, and multiply the denominators together to get the new denominator.
Simplifying Complex Expressions: When simplifying complex expressions involving variables and exponents, it is often helpful to break down the expression into smaller parts, apply the rules of exponents, and then simplify step by step.
Polynomials: A polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables.
