In Exercises 15–24, divide using the quotient rule. x⁹y⁷/x⁴y²

Verified step by step guidance

1. Identify the quotient rule for exponents: $\frac{a^m}{a^n}=a^{m-n}$.
2. Apply the quotient rule to the $x$ terms: $\frac{x^9}{x^4}=x^{9-4}$.
3. Simplify the exponent for $x$: $x^{9-4}=x^5$.
4. Apply the quotient rule to the $y$ terms: $\frac{y^7}{y^2}=y^{7-2}$.
5. Simplify the exponent for $y$: $y^{7-2}=y^5$.

Key Concepts

Quotient Rule

The quotient rule is a fundamental principle in calculus used to differentiate functions that are expressed as the ratio of two other functions. It states that if you have a function f(x) = g(x)/h(x), the derivative f'(x) can be found using the formula f'(x) = (g'(x)h(x) - g(x)h'(x)) / (h(x))². This rule is essential for solving problems involving division of functions.

Simplifying Rational Expressions

Simplifying rational expressions involves reducing fractions to their simplest form by canceling common factors in the numerator and denominator. This process is crucial in algebra as it makes calculations easier and helps in understanding the behavior of functions. For example, in the expression x⁹y⁷/x⁴y², you can simplify by subtracting the exponents of like bases.

Exponent Rules

Exponent rules are a set of mathematical rules that govern the operations of 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 manipulating expressions involving exponents, such as simplifying x⁹y⁷/x⁴y².