Problem 29 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 11-46: Simplify. $$ \left(5 x^{2} y^{-3}\right)\left(4 x^{-5} y^{4}\right) $$

Step-by-Step Solution

Answer

\(\frac{20y}{x^3}\)

Step 1: Distribute the Powers

Multiply the bases that have the same variable by adding their exponents according to the exponent rules. Start with the variable \(x\).

Step 2: Simplify \(x\)

For \(x\), the expression is \(x^{2} \cdot x^{-5}\). To simplify, add the exponents: \[x^{2 + (-5)} = x^{-3}\]

Step 3: Simplify \(y\)

Now for \(y\), the expression is \(y^{-3} \cdot y^{4}\). Again, add the exponents:\[y^{-3 + 4} = y^{1} = y\]

Step 4: Multiply the Coefficients

Multiply the coefficients \(5\) and \(4\):\[5 \times 4 = 20\]

Step 5: Write the Simplified Expression

Combine the simplified coefficients, \(x\) and \(y\) terms into one expression:\[20x^{-3}y\]Finally, rewrite \(x^{-3}\) using positive exponents as \(\frac{1}{x^3}\) to finalize the expression:\[\frac{20y}{x^3}\]

Key Concepts

Understanding Exponent Rules

Exponent rules are essential when you are working with algebraic expressions, especially when dealing with variables raised to various powers. Exponents tell you how many times a number, also called the base, is used in a multiplication. One key exponent rule is that when multiplying like bases, you add their exponents.

For example, if you have the expression

• \( x^a \times x^b \),

the exponents are added together, resulting in \( x^{a+b} \). This is fundamental when simplifying terms that are multiplied together. Also, when dealing with negative exponents, remember that they indicate division rather than multiplication. For instance,

• \( x^{-n} \) can be rewritten as \( \frac{1}{x^n} \).

These rules help keep track of and simplify complex expressions involving exponents.

The Process of Simplification

Simplifying an algebraic expression means reducing it to its most basic form. This often involves using the rules of algebra to combine like terms or simplify individual parts of the expression.

In our exercise, simplification involves:

• Applying the exponent rules to the same variables.
• Then, multiplying out any coefficients as plain numbers, like \(5\) and \(4\) from the equation.

By following these logical steps, the expression becomes easier to understand and use in further calculations. Simplification aims to reduce errors, especially when variables multiply across different parts of a larger equation.

Variable Multiplication Explained

When multiplying variables in algebra, especially those with exponents, it's important to focus on combining like terms carefully. Variables like \(x\) and \(y\) are symbols that represent numbers, and combining them correctly is crucial.

In our problem, we multiply

• \(x^{2} \cdot x^{-5}\) and add their exponents to simplify to \(x^{-3}\),
• as well as \(y^{-3} \cdot y^{4}\) to simplify to \(y^{1}\).

The process involves treating each variable’s exponents separately, using the exponent rules, and then multiplying the coefficients independently. This approach ensures that all parts of the expression are accounted for accurately.

Essentials of Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They can represent general mathematical relationships and can often be simplified or evaluated for given values of the variables.

In algebra, expressions are built from numbers and variables connected by mathematical operations like addition, subtraction, and especially multiplication in this context. For instance, the original expression

• \((5x^2y^{-3})(4x^{-5}y^4)\)

is an expression made up of coefficients \(5\) and \(4\) and variables \(x\) and \(y\) with exponents. Understanding each component's role helps simplify these expressions - turning them into simplified forms like \(\frac{20y}{x^3}\).

This ability to rearrange expressions is foundational, aiding in solving more complex algebraic problems.