Problem 61 - Algebra for College Students

Question

Find each quotient. $$\frac{25 x^{5} y^{6}}{-5 x^{2} y^{4}}$$

Step-by-Step Solution

Answer

The quotient is \( -5x^3y^2 \).

Step 1: Simplify the Coefficients

First, focus on the numerical coefficients in the fraction. The fraction is \( \frac{25}{-5} \), which simplifies to \( -5 \).

Step 2: Simplify the Variables in the Numerator

The numerator \( 25x^5y^6 \) has variables \( x^5 \) and \( y^6 \).

Step 3: Simplify the Variables in the Denominator

The denominator \( -5x^2y^4 \) has variables \( x^2 \) and \( y^4 \).

Step 4: Subtract Exponents of Like Bases

Subtract the exponents of like bases in the numerator and the denominator. For \( x \), subtract the exponent in the denominator from the exponent in the numerator: \( 5 - 2 = 3 \). For \( y \), subtract in the same manner: \( 6 - 4 = 2 \). This gives us \( x^3 \) and \( y^2 \).

Step 5: Express the Simplified Quotient

Combine the simplified coefficient with the simplified variables. The result is \( -5x^3y^2 \).

Key Concepts

Simplifying Fractions

Simplifying fractions is all about making a fraction easier to work with by reducing it to its smallest terms. When working with algebraic fractions, we focus on both the numerical coefficients and the variables within the expression.

• Numerical Coefficient: Start by identifying the numbers in both the numerator and the denominator. Simplify by dividing the numbers. For example, in the fraction \( \frac{25}{-5} \), you would divide 25 by -5, resulting in -5.
• Variable Terms: Look at each variable separately and apply the same simplification process: focusing on exponent rules to reduce the fraction.

Remember that a simplified fraction is one that cannot be reduced any further. This makes further computations easier and the fraction more elegant.

Exponent Rules

Understanding exponent rules is key to simplifying algebraic expressions involving fractions. These rules help you manage powers of variables efficiently.

• Subtraction of Exponents: When you divide variables with the same base, subtract the exponent in the denominator from the exponent in the numerator. For example, if you have \( x^5 \) and \( x^2 \), the result would be \( x^{5-2} \), which simplifies to \( x^3 \).
• Separating Variables: Treat each variable independently. If your expression has several variables, handle each variable according to its own exponents.

The beauty of exponent rules is they allow you to simplify complex expressions neatly and simplify them to forms that are much easier to interpret and use in further calculations.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operators (such as plus or minus signs). These expressions are simplified using the rules and methods of algebra, such as simplifying fractions and applying exponent rules.

• Breaking Down Expressions: Analyze each part of the expression separately – this includes the coefficients, the variable terms, and their respective powers.
• Combining Like Terms: Simplification often involves combining like terms, which means summing or subtracting terms that contain the same variables raised to the same power.
• Simple & Efficient Results: The goal is to rewrite the expression as a minimal number of terms with the lowest possible powers of variables.

Mastering the simplification of algebraic expressions leads to cleaner solutions and prepares you for tackling more complex problems effectively.