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Chapter 6: Problem 36

Factor the given expressions completely. Each is from the technical area indicated. \(\left(\frac{s}{r}\right)^{12}-\left(\frac{s}{r}\right)^{6}\) (molecular interaction)

Short Answer

Expert verified
The expression factors to \(\left(\frac{s}{r}\right)^6 - 1\right)\left(\left(\frac{s}{r}\right)^6 + 1\right).

Step by step solution

01

Recognize the Type of Expression

The expression \(\left(\frac{s}{r}\right)^{12}-\left(\frac{s}{r}\right)^{6}\) is a difference of powers. Specifically, it can be seen as a difference of squares since the powers are even and differ by 6.
02

Apply the Difference of Squares Formula

A difference of squares can be factored using the identity \(a^2-b^2 = (a-b)(a+b)\). Here, identify \(a = \left(\frac{s}{r}\right)^6\) and \(b = 1\). Thus, the expression becomes \(\left( \left(\frac{s}{r}\right)^6 \right)^2 - 1^2 = \left(\frac{s}{r}\right)^{12}-1 = \left(\left(\frac{s}{r}\right)^6 - 1\right)\left(\left(\frac{s}{r}\right)^6 + 1\right)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Difference of Squares
The difference of squares is a common algebraic pattern used in factoring expressions. This occurs when we have two terms that are each perfect squares, and they are separated by a subtraction sign. The formula for factoring a difference of squares is:
  • \[a^2 - b^2 = (a - b)(a + b)\]
This identity simplifies the expression by breaking it down into the product of two binomials. In the original exercise, the expression is \(\left(\frac{s}{r}\right)^{12}-\left(\frac{s}{r}\right)^{6}\). By recognizing that \(a = \left(\frac{s}{r}\right)^6\) and \(b = 1\), the difference of squares can be applied accurately.
Exponents in Algebra
Exponents are a way to represent repeated multiplication of a number. They are important in many areas of algebra. When working with variables, they allow for compact and efficient representation of very large numbers or expressions. For example, \(x^3\) means \(x \times x \times x\). Exponents follow specific rules:
  • Product of Powers: \(a^m \cdot a^n = a^{m+n}\)
  • Power of a Power: \((a^m)^n = a^{m \cdot n}\)
  • Quotient of Powers: \(\frac{a^m}{a^n} = a^{m-n}\)
In the exercise, the concept of exponents is utilized when we identify \(\left(\frac{s}{r}\right)^{12}\) as \(\left(\left(\frac{s}{r}\right)^6\right)^2\). This helps in recognizing the expression as a difference of squares.
Factoring Techniques
Factoring is a critical skill in algebra, allowing us to simplify expressions and solve equations. Beyond the difference of squares, common factoring techniques include:
  • Finding a common factor: Simplifying by distributing a common factor among terms.
  • Trinomials: Factoring expressions in the form of \(ax^2 + bx + c\).
  • Completing the square: Altering an expression to reveal a perfect square trinomial.
In the given exercise, recognizing the expression as a difference of squares allows us to apply specific factoring techniques efficiently, breaking down complex expressions into simpler parts.
Step-by-Step Algebra Solutions
Step-by-step algebra solutions are vital for understanding the process of solving algebraic problems. Breaking down each step clarifies how to approach and manipulate expressions correctly. Here鈥檚 a simple approach.
  • Identify the type of expression: Recognize if it's a difference of squares, quadratic, etc.
  • Apply relevant algebraic identities or formulas: Use knowledge of exponent rules, factoring techniques, and formulas.
  • Simplify the expression: Carefully rewrite the expression in simpler forms.
This methodical approach was used in the solution, where the expression is first identified as a difference of squares, suitable formulas applied, and then the expression is rewritten in its factored form. This clarity in steps helps in building a stronger understanding of algebraic manipulations.

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Most popular questions from this chapter

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