In Exercises 15–24, divide using the quotient rule.50x²y⁷/5xy⁴

Verified step by step guidance

1. Identify the expression to be divided: \( \frac{50x^2y^7}{5xy^4} \).
2. Apply the quotient rule for division of powers: \( \frac{a^m}{a^n} = a^{m-n} \).
3. Divide the coefficients: \( \frac{50}{5} = 10 \).
4. Subtract the exponents for \( x \): \( x^{2-1} = x^1 \).
5. Subtract the exponents for \( y \): \( y^{7-4} = y^3 \).

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. Understanding 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 identifying the behavior of functions. For example, in the expression 50x²y⁷/5xy⁴, you can simplify by dividing both the numerator and denominator by their greatest common factor.

Exponents and Their Properties

Exponents represent repeated multiplication of a base number and have specific rules that govern their manipulation, such as the product of powers, quotient of powers, and power of a power. Understanding these properties is vital when working with algebraic expressions, especially when simplifying terms like x²/x or y⁷/y⁴, as they allow for the correct application of exponent rules to achieve a simplified result.