Problem 43 - Introductory and Intermediate Algebra for College Students 4th

Question

Use the order of operations to simplify each expression. $$(2-6)^{2}-(3-7)^{2}$$

Step-by-Step Solution

Answer

The result of the simplification of the expression (2-6)^{2}-(3-7)^{2} is 0.

Step 1: Compute Inside Parentheses

Calculate the values inside the parentheses first. Thus, the expression becomes: \((2-6)^{2}-(3-7)^{2} = (-4)^{2}-(-4)^{2}\).

Step 2: Squaring the Numbers

Next, square the numbers to get: \((-4)^{2}-(-4)^{2} = 16 - 16 \).

Step 3: Subtract the Results

Finally, subtract the two squared numbers. Therefore, the simplified expression is: \(16 - 16 = 0 \).

Key Concepts

Simplify Expressions

When it comes to simplifying algebraic expressions, understanding the order of operations is essential. This set of rules guides us through which steps to tackle first so we can reduce expressions down to their simplest form without error.

The order of operations, often remembered by the acronym PEMDAS, stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). To avoid confusion, always start with calculations inside parentheses, followed by exponents, and work your way down through multiplication, division, addition, and subtraction.

In our exercise, we first deal with the parentheses to identify the actual numbers we will be working with before proceeding to the next steps. Simplifying expressions in the correct order guarantees accuracy and is a fundamental skill in mathematics that not only helps in homework but also in solving real-world problems.

Parentheses in Math

Parentheses are not just curvy brackets that look nice around numbers; they play a crucial role in mathematical calculations. These symbols tell us to process what’s within them before moving on to any other part of the expression. Think of them as a polite request from the numbers inside, asking to be dealt with first.

In the given example, the parentheses hold the subtraction operations, which must be completed prior to squaring the numbers. The use of parentheses ensures that we first find the difference between the numbers inside them, which is \(2-6\) resulting in \-4\ and \(3-7\) also resulting in \-4\.

Without the correct use of parentheses, expressions could be calculated incorrectly, leading to the wrong solution. Hence, they are not just important but necessary to correctly interpret and solve mathematical expressions.

Squared Numbers

A squared number is simply a number multiplied by itself. The notation for squaring a number is a small '2' written just above and to the right of the number, known as an exponent. For instance, \(3^2\) translates to \(3\times 3\), which equals 9.

In our example, we square negative four, written as \(\left(-4\right)^2\), which gives us 16. Remember, the square of a negative number is always positive because multiplying two negative numbers together yields a positive result. After squaring both instances of \-4\ in the expression, we get \(16-16\), which simply equals 0. Squaring numbers is a fundamental operation and understanding how it works enhances your ability to simplify expressions and solve more complex mathematical problems.