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Chapter 2: Problem 93

Simplify each of the following. $$(6+2)(6-2)$$

Short Answer

Expert verified
The simplified form of \((6+2)(6-2)\) is 32.

Step by step solution

01

Recognize the Form

The expression \((6+2)(6-2)\) is in the form of a difference of squares. This form looks like \((a+b)(a-b)\), which simplifies to \(a^2-b^2.\) Recognizing this pattern helps to simplify the expression quickly.
02

Identify Components

Identify \(a\) and \(b\) from the expression. Here, \(a = 6\) and \(b = 2.\)
03

Apply the Difference of Squares Formula

Apply the formula for the difference of squares: \(a^2 - b^2.\) Substitute the identified values: \(6^2 - 2^2.\) This simplifies to \(36 - 4.\)
04

Perform the Subtraction

Calculate \(36 - 4 = 32.\) So, the simplified form of \((6+2)(6-2)\) is \(32.\)

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Key Concepts

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

Simplifying Expressions
Simplifying expressions is an essential skill in mathematics. It means reducing an expression to its simplest form. For example, with the expression \((6+2)(6-2)\), you can simplify it by recognizing a special pattern known as the difference of squares. Once you see these patterns, calculations become much faster and simpler. In this case, you simplify the expression to just a single number using the steps:
  • Recognize the pattern form \((a+b)(a-b)\)
  • Identify the parts: \(a\) and \(b\)
  • Use the formula \(a^2-b^2\).
By substituting and performing basic arithmetic operations, the complex expression becomes \(32\).
Polynomials
Polynomials are expressions that can have constants, variables, and exponents. They don't have division by a variable. The expression \((6+2)(6-2)\) is a simple polynomial because it involves addition and multiplication without any division or complex variables. Understanding polynomials is essential:
  • They form the basis of more complex algebraic equations.
  • Recognize common forms like the difference of squares to ease simplification.
  • Allows for substitution and manipulation through algebraic properties.
Specifically, using the difference of squares helps in simplifying polynomial expressions quickly by transforming them to their simplest terms.
Prealgebra Concepts
Prealgebra forms the foundation for all higher-level math. Concepts like recognizing patterns in polynomials are core to prealgebra. When you see expressions like \((6+2)(6-2)\), you're applying prealgebra concepts to identify easier computation methods.
  • Identify patterns such as the difference of squares.
  • Place numbers into recognized patterns to simplify calculations.
  • Perform basic arithmetic for quick solutions.
Understanding prealgebra skills such as recognizing patterns and simplifying expressions ensures a smooth transition to more advanced algebraic operations and polynomials.

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Most popular questions from this chapter

The Eiffel Tower has several levels visitors can walk around on. The first is 57 meters above the ground, the second is 115 meters high, and the third level is 276 meters high. What is the location of someone standing on the first level in relation to someone standing on the third level?

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Subtract. $$156-(-243)$$

Simplify as much as possible by first changing all subtractions to addition of the opposite and then adding left to right. $$-300+600-(-200)$$
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