Problem 77 - Algebra 2

Question

PREREQUISITE SKILL. Simplify. Assume that no variable equals 0. $$ \frac{x^{2} y^{3}}{x y} $$

Step-by-Step Solution

Answer

The simplified expression is \( xy^2 \).

Step 1: Identify the Expression

We start with the given expression \( \frac{x^2 y^3}{x y} \). Our goal is to simplify this expression.

Step 2: Simplify the Numerator and Denominator separately

The numerator is \( x^2 y^3 \) and the denominator is \( x y \). Each component will be simplified independently if possible.

Step 3: Apply the Division Rule for Exponents

The division rule for exponents states that \( \frac{a^m}{a^n} = a^{m-n} \). Apply this to \( \frac{x^2}{x} \) to get \( x^{2-1} = x^1 = x \) and \( \frac{y^3}{y} \) to get \( y^{3-1} = y^2 \).

Step 4: Write the Simplified Expression

After applying the division rule, the remaining expression is \( xy^2 \).

Step 5: Consider Variables Cannot Be Zero

The problem states that no variable equals zero, so our solution must assume that neither \( x \) nor \( y \) is zero.

Key Concepts

Division rule for exponents

When dealing with algebraic expressions, especially those that involve exponents, you'll often need to simplify terms using the division rule for exponents. This rule is an essential tool in algebra, enabling you to reduce expressions for ease of use. The division rule states that when you divide two like bases with different exponents, you simply subtract the exponent of the denominator from the exponent of the numerator. For example, consider the expression \( \frac{x^a}{x^b} \). Using the division rule, it simplifies to \( x^{a-b} \). This concept helps simplify and break down complex algebraic fractions, making them easier to manage and work with. It’s crucial to remember this rule applies only when the bases are the same, offering a streamlined way to handle expressions that might otherwise look intimidating.

Prerequisite skills in algebra

Algebra can sometimes seem complex, but understanding it begins with mastering a few essential skills. These foundational skills will make you comfortable handling algebraic operations and transitions.

• Understanding variables and constants: Recognize what they represent in different scenarios.
• Basic operations: Become proficient in addition, subtraction, multiplication, and division of variables and constants.
• Exponent rules: Know how to apply them when simplifying expressions.
• Factoring expressions: Break down complex expressions into simpler components effectively.

Mastering these basic skills will ensure you have the necessary tools to tackle more advanced topics like algebraic fractions and expressions. These foundations pave the way for smoother experiences when solving algebra problems.

Algebraic fractions

Algebraic fractions are expressions that include polynomials in the numerator, the denominator, or both. Just like numerical fractions, simplifying algebraic fractions involves similar processes. To simplify an algebraic fraction, first, identify if any terms in the numerator and denominator can be reduced. This might involve factoring first to see if any like terms exist that could cancel out. Once that’s done, apply any algebraic rules like the division rule for exponents or combining like terms to make the fraction simpler. It’s crucial to remember that the simpler form of the fraction should still convey the same mathematical meaning as the original. Understanding how to handle algebraic fractions is a key part in solving more complex algebra problems.