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.) [(y² +2)⁵(3y) - y³(6)(y²+2)⁴(3y)] / [(y²+2)⁷]

Verified step by step guidance

1. Start by writing the given expression clearly: \(\frac{(y^2 + 2)^5 (3y) - y^3 (6) (y^2 + 2)^4 (3y)}{(y^2 + 2)^7}\)
2. Look for common factors in the numerator. Notice that both terms contain \((y^2 + 2)^4\) and \$3y\(. Factor these out: \)(y^2 + 2)^4 (3y) \left[(y^2 + 2) - 6 y^3 \right]$
3. Rewrite the numerator using the factored form: \((y^2 + 2)^4 (3y) \left[(y^2 + 2) - 6 y^3 \right]\)
4. Now substitute the factored numerator back into the original expression: \(\frac{(y^2 + 2)^4 (3y) \left[(y^2 + 2) - 6 y^3 \right]}{(y^2 + 2)^7}\)
5. Simplify the expression by dividing powers of \((y^2 + 2)\) in numerator and denominator using the law of exponents: \(\frac{(y^2 + 2)^4}{(y^2 + 2)^7} = (y^2 + 2)^{4 - 7} = (y^2 + 2)^{-3}\). So the expression becomes: \$3y (y^2 + 2)^{-3} \left[(y^2 + 2) - 6 y^3 \right]$

Key Concepts

Factoring Expressions

Factoring involves rewriting expressions as products of simpler expressions. It helps identify common factors in the numerator and denominator, which can be canceled to simplify rational expressions. For example, recognizing powers of (y^2 + 2) and common terms like 3y allows easier simplification.

Properties of Exponents

Understanding exponent rules is essential when simplifying expressions with powers. For instance, when multiplying terms with the same base, add exponents; when dividing, subtract exponents. This helps simplify terms like (y^2 + 2)^5 and (y^2 + 2)^4 in the numerator and denominator.

Simplifying Rational Expressions

A rational expression is a fraction where numerator and denominator are polynomials. Simplifying involves factoring, canceling common factors, and reducing the expression to its simplest form. Assuming variables represent positive real numbers ensures no issues with domain restrictions during simplification.