Problem 45 - Prealgebra

Question

Use any of the rules developed in this chapter and the rule for order of operations to simplify each of the following expressions as much as possible. [Examples 6–9] $$100 \div(-5)^{2}$$

Step-by-Step Solution

Answer

The expression simplifies to 4.

Step 1: Identify the Order of Operations

In order to simplify the given expression \(100 \div(-5)^{2}\), we need to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).

Step 2: Calculate the Exponent

First, we calculate the exponent \((-5)^2\). Squaring \(-5\) means multiplying it by itself: \((-5) \times (-5) = 25\).

Step 3: Perform the Division

Next, divide 100 by the result of the exponent calculation. So, the expression becomes \(100 \div 25\). Calculating this gives us 4.

Key Concepts

Understanding PEMDAS

Simplifying mathematical expressions requires a clear understanding of the Order of Operations, which can be easily remembered using the acronym **PEMDAS**. This rule is crucial because it tells us the order in which to perform operations to accurately evaluate an expression. The acronym stands for:

• **P**arentheses: Solve anything in parentheses first.
• **E**xponents: Next, calculate powers and roots.
• **M**ultiplication and **D**ivision: These operations are on the same level and should be completed from left to right.
• **A**ddition and **S**ubtraction: Like multiplication and division, these should also be performed from left to right.

By following PEMDAS, you ensure your calculations are accurate. In the example given, the expression is in the form of a division and exponent. To solve it correctly, understand that exponents come before division.

All About Exponents

Exponents are a way to express repeated multiplication. The exponent tells us how many times to multiply the base number by itself. For instance, \((-5)^2\) means that you will multiply \(-5\) by itself, resulting in \((-5) \times (-5) = 25\).
This might seem simple, but paying attention is key when handling negative numbers. When we square \(-5\), the negative sign is part of the base, so we really multiply \(-5\) with \(-5\) giving you a positive result. Remember, if there is no parenthesis and it's written as \(-5^2\), only \(5\) has the exponent, and the negative sign is not squared.
Exponents can change the value rapidly, and understanding them is vital, especially when simplifying expressions.

Simplifying Expressions Made Easy

When faced with simplifying expressions, start by analyzing the expression to identify which operations to perform first, according to PEMDAS. Take our example \(100 \div (-5)^2\):

• We begin with exponents, \((-5)^2\), to get \(25\).
• Then, shift focus to the remaining operation, which is the division: \(100 \div 25\).
• Perform the division to simplify the expression to \(4\).

The aim of simplification is to express the expression in its most straightforward form. This practice not only makes solving problems easier but also reduces errors. Always follow the sequence, and if in doubt, break it down step-by-step by highlighting each operation.