Simplify each exponential expression in Exercises 23–64. $\frac{10x^4 y^9}{30x^{12} y^{-3}}$

Verified step by step guidance

1. Start by writing the expression clearly: \(\frac{10x^{4} y^{9}}{30x^{12} y^{-3}}\).
2. Simplify the coefficients (numerical parts) by dividing 10 by 30, which gives \(\frac{10}{30}\).
3. Apply the quotient rule for exponents to the variables with the same base: for \(x\), subtract the exponents in the denominator from those in the numerator: \(x^{4 - 12}\); for \(y\), do the same: \(y^{9 - (-3)}\).
4. Rewrite the expression with the simplified coefficients and the new exponents: \(\frac{10}{30} x^{4 - 12} y^{9 - (-3)}\).
5. Simplify the numerical fraction and the exponents to get the final simplified expression.

Key Concepts

Laws of Exponents

The laws of exponents govern how to simplify expressions involving powers. Key rules include dividing powers with the same base by subtracting exponents, multiplying powers by adding exponents, and handling negative exponents by rewriting them as reciprocals. These rules allow simplification of expressions like the given one.

Simplifying Algebraic Fractions

Simplifying algebraic fractions involves reducing coefficients and variables separately. Coefficients are simplified by dividing their numerical values, while variables with exponents are simplified using exponent rules. This process helps to rewrite the expression in its simplest form.

Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the positive exponent. For example, x^(-n) equals 1/x^n. Understanding this concept is essential to simplify terms like y^(-3) in the denominator by moving them to the numerator with a positive exponent.