Problem 10 - Elementary Algebra

Question

Perform each multiplication in one step. $$ (x-y)^{3} \cdot 4(x-y)^{2} $$

Step-by-Step Solution

Answer

Question: Perform the given multiplication in one step: $(x-y)^{3} \cdot 4(x-y)^{2}$. Answer: $4(x-y)^{5}$

Step 1: Identify the given expression

The given expression is: $$ (x-y)^{3} \cdot 4(x-y)^{2} $$

Step 2: Apply the laws of exponents

When multiplying two terms with the same base, we can add their exponents. In this case, apply the exponent law \((a^n)(a^m) = a^{n+m}\) to the given expression: $$ (x-y)^{3} \cdot 4(x-y)^{2} = 4(x-y)^{3+2} $$

Step 3: Simplify the expression

Now, simplify the expression by combining the exponents: $$ 4(x-y)^{5} $$ The multiplication has been performed, and the simplified expression is \(4(x-y)^{5}\).

Key Concepts

Multiplication of Expressions

In algebra, multiplication of expressions involves combining two or more algebraic terms. Think of it as a way to bring two or more expressions together into one neat package. When multiplying expressions, we look for common terms to combine.
Consider the exercise: \((x-y)^{3} \cdot 4(x-y)^{2}\). Here, we have two expressions: \((x-y)^{3}\) and \(4(x-y)^{2}\). Both expressions share a common base: \((x-y)\). Our goal is to multiply these expressions by essentially considering the coefficients and the terms separately. Therefore, we:

• Multiply the coefficients: 1 and 4. Here, the coefficient is 1 in \((x-y)^{3}\) because there's no explicit number before the expression.
• Multiply the variable parts, using exponent rules.

Bringing it all together, you multiply the numbers and then apply the exponent rule to end with a simplified expression.

Laws of Exponents

The laws of exponents are critical in simplifying expressions where exponential terms play a role. These rules dictate how to perform operations like multiplication, division, and power to a power all involving exponents.
One essential law of exponents is the product of powers rule:

• If you have \(a^n \cdot a^m\), it simplifies to \(a^{n+m}\).

This rule says that when multiplying terms with the same base, simply add their exponents together. In the exercise \((x-y)^{3} \cdot 4(x-y)^{2}\), both terms have the base \((x-y)\).
By applying the product of powers rule, it's clear the expression simplifies to \((x-y)^{3+2}\), which becomes \((x-y)^{5}\). Thus, by understanding the laws of exponents, you can make quick work of what might seem initially complex.

Simplification of Algebraic Expressions

Simplification is the process of making an algebraic expression as concise and efficient as possible without altering its value. This often involves combining like terms using arithmetic operations and applying rules like those of exponents.
For the expression we are working with, the original problem statement has \((x-y)^{3} \cdot 4(x-y)^{2}\). First, we apply the laws of exponents to combine the terms. This step, which you've seen from the rules of exponents, gives us \((x-y)^{5}\).

• Next, ensure that any coefficients are maintained with the simplified expressions. Hence, the final simplified form includes the coefficient 4 and the new exponent, resulting in \(4(x-y)^{5}\).

Proper simplification ensures the expression is as straightforward as possible, making further calculations or analysis easier and more efficient. Being able to identify and combine powers effectively is key in this process.