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.) [10(4x²-9)² - 25x(4x²-9)³] / [15(4x²-9)⁶]

Verified step by step guidance

1. Identify the common factors in the numerator and denominator. Notice that both contain powers of the expression \((4x^2 - 9)\).
2. Rewrite the numerator by factoring out the smallest power of \((4x^2 - 9)\) present in both terms. The numerator is \$10(4x^2 - 9)^2 - 25x(4x^2 - 9)^3\(, so factor out \)(4x^2 - 9)^2$:
3. \[ (4x^2 - 9)^2 (10 - 25x(4x^2 - 9)) \]
4. Simplify inside the parentheses: \$10 - 25x(4x^2 - 9)$.
5. Rewrite the entire expression as a fraction, then divide numerator and denominator by the common factor \((4x^2 - 9)^2\) to reduce the powers in the denominator:
6. \[ \frac{(4x^2 - 9)^2 (10 - 25x(4x^2 - 9))}{15(4x^2 - 9)^6} = \frac{10 - 25x(4x^2 - 9)}{15(4x^2 - 9)^{6-2}} = \frac{10 - 25x(4x^2 - 9)}{15(4x^2 - 9)^4} \]

Key Concepts

Factoring Polynomials

Factoring involves rewriting a polynomial as a product of simpler polynomials or expressions. Recognizing common patterns like difference of squares or factoring out the greatest common factor helps simplify complex expressions. In this problem, factoring terms like (4x^2 - 9) is essential to reduce the rational expression.

Simplifying Rational Expressions

Simplifying rational expressions means reducing fractions by canceling common factors in the numerator and denominator. This requires factoring both parts completely and then dividing out shared factors. It is important to consider the domain restrictions, especially when variables represent positive real numbers.

Exponent Rules for Expressions

Exponent rules govern how to handle powers of expressions, such as multiplying powers with the same base by adding exponents or factoring powers out of terms. Understanding how to manipulate expressions like (4x^2 - 9)^n is crucial for combining and simplifying terms in the numerator and denominator.