Problem 81 - Algebra 1

Question

Evaluate the expression. $$\frac{(3-6)^{2}+6}{-5}$$

Step-by-Step Solution

Answer

-3

Step 1: Solve the operation within parentheses

We'll first handle the operations in parentheses, that is, \( (3-6) \). Subtract 6 from 3, to get \( -3 \).

Step 2: Square the result of Step 1

Square the result obtained from step 1, which is \(-3\), to get \(9.\)

Step 3: Add 6 to the squaring result of Step 2

Add 6 to the value obtained from step 2, \(9\). The result is \(9 + 6 = 15\).

Step 4: Divide the result in Step 3 by -5

Next, we need to divide the result from step 3, \(15\), by \(-5\). The result is \(-3\)

Key Concepts

Order of Operations

Understanding the order of operations is fundamental in evaluating algebraic expressions correctly. It's a set of rules that dictates the sequence in which different operations are to be performed to ensure consistent results. The conventional mnemonic 'Please Excuse My Dear Aunt Sally' represents these rules in the sequence: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This mnemonic translates to the acronym PEMDAS.

To apply these rules to the original exercise, we start by solving the expression within the parentheses \( (3-6) \), before addressing the exponent \( ^2 \). Only after these operations can we move on to addition and then the final division. By adhering to these guidelines strictly, we achieve the correct answer without ambiguity.

Simplifying Expressions

Simplifying expressions is a way of rewriting them in the most basic form without changing their value. The aim is to make an expression easier to understand and work with. In our given example, the simplification process involves reducing an expression to fewest terms.

After handling the parentheses and exponents, the expression \( (3-6)^2 + 6 \) is simplified to \( 9 + 6 \), which further simplifies to \( 15\). The goal is to systematically reduce the complexity of the expression, while preserving its mathematical integrity. It's important when simplifying to combine like terms accurately and to pay close attention to the signs of the numbers involved to prevent errors.

Arithmetic Operations

Arithmetic operations are the building blocks of mathematics. They include addition, subtraction, multiplication, and division. In algebra, these operations work just like they do with plain numbers, but we often apply them to expressions that can contain variables, exponents, and parentheses.

In the final step of our exercise, we employed a basic arithmetic operation—division. We divided the sum \( 15 \) by \( -5 \) to arrive at the final answer, \( -3 \). Arithmetic operations can be influenced by the hierarchy outlined in the order of operations, and they must be accurately executed to ensure the expression is correctly evaluated. A careful consideration of the signs (+/-) is especially crucial to avoid common mistakes in calculation.