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Chapter 5: Problem 124

Simplify. $$ \frac{2(8-3)-6+2}{3^{2}-2^{3}} $$

Short Answer

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Step by step solution

01

鈥� Simplify the Numerator

Start with the numerator: \(2(8-3)-6+2\). First, calculate \(8-3\) to get 5. Then multiply this result by 2: \(2 \times 5 = 10\). Finally, calculate \(10-6+2\), which gives 6.
02

鈥� Simplify the Denominator

Next, simplify the denominator: \(3^{2}-2^{3}\). Calculate \(3^{2}\) to get 9, and \(2^{3}\) to get 8. Then subtract these results: \(9-8=1\).
03

鈥� Divide the Simplified Numerator by the Simplified Denominator

Now divide the simplified numerator by the simplified denominator: \(\frac{6}{1} = 6\).

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Key Concepts

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

Order of Operations
Understanding the order of operations is crucial when simplifying algebraic expressions. It dictates the sequence in which you should solve parts of an expression. The common acronym to remember this is PEMDAS:

  • **P**arentheses
  • **E**xponents
  • **M**ultiplication
  • **D**ivision
  • **A**ddition
  • **S**ubtraction

For our exercise, we address each step carefully. First, the expression inside parentheses \(8-3\) is simplified to \(5\). Next, any exponents are handled, such as \(3^2\) and \(2^3\). Multiplications and divisions are handled from left to right, followed by addition and subtraction.
Numerator and Denominator
An algebraic fraction has two main parts: the numerator (top part) and the denominator (bottom part). Simplifying these separately helps manage the complexity. In the given problem, the numerator is \(2(8-3)-6+2\) and the denominator is \(3^2 - 2^3\).

To handle the numerator:
  • Simplify the expression inside the parentheses first: \(8-3 = 5\).
  • Multiply the result by 2: \(2 \times 5 = 10\).
  • Then perform the addition and subtraction from left to right: \(10-6+2 = 6\).

For the denominator:
  • Compute \(3^2 = 9\).
  • Compute \(2^3 = 8\)
  • Subtract the results: \(9-8 = 1\).
Exponents
Exponents are a shorthand way to show that a number is multiplied by itself a certain number of times. In our example, we have two exponents: \(3^2\) and \(2^3\).

  • \(3^2\) means multiplying 3 by itself: \(3 \times 3 = 9\).
  • \(2^3\) means multiplying 2 by itself three times: \(2 \times 2 \times 2 = 8\).

Exponents must be evaluated before other operations like addition, subtraction, and multiplication according to the order of operations (PEMDAS). Utilizing exponents helps simplify expressions and solve them accurately.

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