Simplify each exponential expression in Exercises 23–64. 10x^4 y^9/30x^12 y^−3

Exponential rules govern how to manipulate expressions involving exponents. Key rules include the product of powers (a^m * a^n = a^(m+n)), the quotient of powers (a^m / a^n = a^(m-n)), and the power of a power ( (a^m)^n = a^(m*n)). Understanding these rules is essential for simplifying expressions with variables raised to powers.

Simplifying fractions involves reducing the numerator and denominator by their greatest common factor (GCF). In algebraic expressions, this means factoring out common terms and canceling them. This process is crucial for making complex expressions more manageable and easier to interpret.

Negative exponents indicate the reciprocal of the base raised to the opposite positive exponent (a^(-n) = 1/a^n). This concept is important when simplifying expressions, as it allows for the transformation of terms in the denominator into terms in the numerator, facilitating easier simplification of the overall expression.