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Chapter 1: Problem 51

Find the value of each expression. $$ 3^{2}[(11+3)-4] $$

Short Answer

Expert verified
The value of the expression is 90.

Step by step solution

01

Identify the expression inside the brackets

The given expression is \(3^{2}[(11 + 3) - 4]\). Focus initially on the expression inside the square brackets: \((11 + 3) - 4\).
02

Simplify the addition inside the brackets

Calculate the sum inside the brackets: \(11 + 3 = 14\), so the expression becomes \(3^{2}[14 - 4]\).
03

Simplify the subtraction inside the brackets

Next, calculate the difference: \(14 - 4 = 10\). Now, the expression is \(3^{2}[10]\).
04

Simplify the exponentiation

Calculate the value of \(3^{2}\), which is \(3 \times 3 = 9\). Now the expression is \(9 \times 10\).
05

Final multiplication

Multiply the results: \(9 \times 10 = 90\).

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Key Concepts

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

Evaluating Expressions
When you are dealing with algebraic expressions, it is important to follow certain rules to evaluate them correctly. Evaluating expressions involves finding their values by performing the arithmetic operations in the correct order. In the given exercise, the expression is \(3^{2}[(11 + 3) - 4]\).

We start by focusing on the innermost part of the expression, which is within the brackets. Here, \(11 + 3 - 4\) needs simplification step by step. We follow the order of operations to do this.

The terms within the brackets are computed first, then exponentiation, and finally, any remaining arithmetic operations are performed.
Exponentiation
Exponentiation is a mathematical operation involving two numbers: a base and an exponent. It is represented as \(a^b\), where \(a\) is the base and \(b\) is the exponent. In simpler terms, it means multiplying the base \(a\) by itself \(b\) times.

For example, in \(3^2\), the base is 3 and the exponent is 2, meaning we multiply 3 by itself: \(3 \times 3 = 9\).

In our expression, we come across \(3^{2}\). So, we first calculate the exponentiation step, which simplifies to 9. Then, the expression becomes \(9[10]\).
Arithmetic Operations
Arithmetic operations consist of addition, subtraction, multiplication, and division. These operations are performed following a specific order known as the order of operations. The order of operations is often remembered by the acronym PEMDAS:

  • Parentheses
  • Exponents
  • Multiplication and Division (left to right)
  • Addition and Subtraction (left to right)

In the given problem \(3^{2}[(11 + 3) - 4]\), we follow these rules to solve it step by step. First, compute inside the parentheses: \(11 + 3 = 14\). Then, we perform the subtraction: \(14 - 4 = 10\).

After simplifying inside the brackets, we handle the exponentiation: \(3^{2} = 9\). Finally, we perform the multiplication: \(9 \times 10 = 90\). So, the expression evaluates to 90.

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