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 given rational expression: \(\frac{2(2x-3)^{1/3} - (x-1)(2x-3)^{-2/3}}{(2x-2)^{-2/3}}\).
2. Factor expressions where possible. Notice that \$2x-2\( can be factored as \)2(x-1)\(, so rewrite the denominator as \)(2(x-1))^{-2/3}$.
3. Look for common factors in the numerator. Both terms contain powers of \((2x-3)\): \( (2x-3)^{1/3} \) and \( (2x-3)^{-2/3} \). Factor out the smaller power, which is \( (2x-3)^{-2/3} \), from the numerator.
4. After factoring, the numerator becomes \( (2x-3)^{-2/3} [2(2x-3)^{(1/3 + 2/3)} - (x-1)] = (2x-3)^{-2/3} [2(2x-3)^{1} - (x-1)] \).
5. Rewrite the entire expression as \( \frac{(2x-3)^{-2/3} [2(2x-3) - (x-1)]}{(2(x-1))^{-2/3}} \). Then, use the property of exponents to rewrite the division as multiplication by the reciprocal, and simplify the expression inside the brackets.

Key Concepts

Rational Expressions

Rational expressions are fractions where the numerator and/or denominator are polynomials or algebraic expressions. Simplifying them involves factoring, canceling common factors, and applying algebraic rules to rewrite the expression in a simpler form.

Exponents and Radicals with Rational Powers

Rational exponents represent roots and powers simultaneously, such as x^(m/n) meaning the n-th root of x raised to the m-th power. Understanding how to manipulate expressions with rational exponents, including negative exponents, is essential for simplification.

Factoring and Canceling Common Factors

Factoring involves rewriting expressions as products of simpler expressions. Identifying and canceling common factors in the numerator and denominator reduces the expression to its simplest form, which is a key step in simplifying rational expressions.