Simplify each rational expression. Assume all variable expressions represent positive real numbers. (Hint: Use factoring and divide out any common factors as a first step.) [2(2x-3)^1/3 - (x-1)(2x-3)^-2/3] / [(2x-2)^-2/3]

Verified step by step guidance

1. Identify the expression to simplify: $\frac{2(2x-3{)}^{1/3}-(x-1)(2x-3{)}^{-2/3}}{(2x-2{)}^{-2/3}}$.
2. Factor out the common term $(2x-3{)}^{-2/3}$ from the numerator: $(2x-3{)}^{-2/3}[2(2x-3{)}^1-(x-1)]$.
3. Simplify the expression inside the brackets: $2(2x-3)-(x-1)$.
4. Simplify the denominator: $(2x-2{)}^{-2/3}$ can be rewritten as $(2(x-1){)}^{-2/3}$.
5. Divide the factored numerator by the simplified denominator and simplify further if possible.

Key Concepts

Rational Expressions

Rational expressions are fractions where the numerator and denominator are polynomials. Understanding how to manipulate these expressions, including simplifying, factoring, and finding common factors, is essential for solving problems involving them. In this context, recognizing the structure of the expression allows for effective simplification.

Factoring

Factoring involves breaking down a polynomial into simpler components, or factors, that can be multiplied together to obtain the original polynomial. This process is crucial in simplifying rational expressions, as it helps identify and eliminate common factors in the numerator and denominator, leading to a more manageable form of the expression.

Exponents and Negative Exponents

Exponents indicate how many times a number is multiplied by itself, while negative exponents represent the reciprocal of the base raised to the corresponding positive exponent. Understanding how to manipulate expressions with exponents, especially negative ones, is vital for simplifying rational expressions, as it allows for the conversion of terms into a more usable form.