Problem 32 - Algebra and Trigonometry

Question

Simplify each expression. $$ \frac{[2(r-s)]^{2}}{(r-s)^{3}} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{4}{r-s}\).

Step 1: Expand the Numerator

The numerator of the expression is \([2(r-s)]^2\). To expand this, apply the power to each component: \[[2(r-s)]^2 = (2)^2 imes (r-s)^2 = 4(r-s)^2\] Thus, the numerator becomes \(4(r-s)^2\).

Step 2: Rewrite the Expression

Using the expanded numerator, rewrite the entire expression: \[\frac{4(r-s)^2}{(r-s)^3}\]

Step 3: Simplify the Expression by Canceling Out Common Terms

Observe that \((r-s)^2\) in the numerator and \((r-s)^3\) in the denominator have a common factor of \((r-s)^2\). By dividing both by \((r-s)^2\), you simplify the expression:\[\frac{4(r-s)^2}{(r-s)^3} = \frac{4}{r-s}\]

Step 4: Final Simplified Expression

After canceling out \((r-s)^2\) from both the numerator and the denominator, the expression simplifies to \(\frac{4}{r-s}\). This is the simplest form of the given expression.

Key Concepts

Rational Expressions

Rational expressions are important components in algebra that involve fractions where the numerator and the denominator are polynomial expressions. Understanding how to manipulate and simplify these expressions is crucial when solving algebra problems. When working with rational expressions, the key steps often involve:

• Simplifying polynomials in the numerator and the denominator.
• Identifying and canceling common factors between the numerator and the denominator.
• Ensuring the expression is in its simplest form.

In the given exercise, we are tasked with simplifying a rational expression involving powers of polynomials. The expression \( \frac{[2(r-s)]^{2}}{(r-s)^{3}} \) features a polynomial \((r-s)\) raised to different exponents in both the numerator and the denominator. Mastering the technique to address such expressions enhances your ability to tackle more complex algebraic equations effectively.

Exponent Rules

Exponent rules are the guidelines that help us simplify expressions involving powers of the same base. These rules are critical in algebra, especially when dealing with rational expressions. Here are a few key exponent rules relevant to this context:

• Product of Powers Rule: \(a^m \times a^n = a^{m+n}\). This rule implies multiplying powers with the same base by adding their exponents.
• Power of a Power Rule: \((a^m)^n = a^{m \times n}\). This means when raising a power to another power, multiply the exponents.
• Quotient of Powers Rule: \(\frac{a^m}{a^n} = a^{m-n}\). This rule allows for the subtraction of the exponents when dividing powers with the same base.

In the step-by-step solution, these rules are applied to successfully simplify the rational expression by addressing the powers of \((r-s)\). Specifically, the first step involves using the power of a power rule to expand \([2(r-s)]^2\) to \(4(r-s)^2\). Understanding these rules will strengthen your algebraic skills.

Simplification Steps

Simplification processes in algebra are about making expressions more manageable, while preserving their equivalence. The steps provided in the solution outline a clear path to achieve this. Here's a recap:

• Expand the Numerator: By applying the power rule, the expression \([2(r-s)]^2\) becomes expanded into \(4(r-s)^2\).
• Identify and Cancel Common Factors: Look at both the numerator and the denominator for common factors. Here, the factor \((r-s)^2\) appears in both parts.
• Simplify: Cancel the common factor \((r-s)^2\) to reduce the expression to its simplest form, \(\frac{4}{r-s}\).

It is important to meticulously follow these simplification steps, ensuring both accuracy and efficiency when handling rational expressions. This practice not only helps in homework assignments but also enhances problem-solving abilities in various mathematical contexts.