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Chapter 1: Problem 45

Simplify the given expression. $$ 6-5[11-(2+8)] $$

Short Answer

Expert verified
The simplified expression is 1.

Step by step solution

01

Simplify Inside the Innermost Parentheses

Start by solving the expression inside the innermost parentheses: \( (2+8) \). Compute \( 2+8 \) to get \( 10 \).
02

Solve the Expression Inside the Brackets

Replace \( (2+8) \) with \( 10 \) in the expression, which gives: \( 6 - 5[11 - 10] \). Next, compute \( 11-10 \) to get \( 1 \).
03

Multiply by the Coefficient Outside the Brackets

Now substitute \( 1 \) into the expression: \( 6 - 5 imes 1 \). Compute \( 5 imes 1 \) to get \( 5 \).
04

Final Simplification

Subtract the product obtained in the previous step from 6: \( 6 - 5 = 1 \).

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Key Concepts

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

Simplification
Simplification is a vital skill in algebra that involves reducing expressions to their simplest form. It can make complex problems easier to work with and solve. When simplifying, always follow specific steps rather than rushing to the answer, as this can prevent errors.In our example, the original expression is \( 6 - 5[11 - (2+8)] \). First, solve the innermost operation, the parentheses within the brackets. This structured approach ensures you handle all operations systematically and accurately.Here's why simplification helps:
  • Reduces complex expressions, making them easier to understand and solve.
  • Helps identify and eliminate unnecessary steps or components in math problems.
  • Provides clarity in the solution process by following the order of operations.
Parentheses
Parentheses are crucial in mathematical expressions because they determine the order of operations. When you see an expression with parentheses, always solve what's inside them first.In the exercise, we first addressed the expression within the parentheses \((2+8)\), which simplifies to \(10\). Solving inside the parentheses ensures you respect the hierarchy of operations, which is fundamental when calculating correctly.Further points to consider:
  • Parentheses override the usual precedence rules and must be dealt with first.
  • Nesting of parentheses requires solving from the innermost outward.
  • Always double-check after simplifying expressions within parentheses to ensure accuracy.
Arithmetic Expressions
Arithmetic expressions consist of numbers, operators, and sometimes parentheses, which together convey a value that needs calculus. In our original problem \( 6 - 5[11 - (2+8)] \), each component communicates a specific part of the expression.To evaluate correctly, understand each element of an arithmetic expression:
  • Numbers are the basic units with specific values.
  • Operators such as addition, subtraction, multiplication, and division determine how these numbers relate to one another.
  • Combined with parentheses or brackets, they form structured expressions, dictating the computational order.
Being comfortable with arithmetic expressions allows for fluid manipulation and simplification, ensuring correct solutions to problems.

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