Problem 38 - Introductory Algebra for College Students

Question

In Exercises \(29-72,\) use the order of operations to simplify each expression. $$10^{2}-100 \div 5^{2} \cdot 2-1$$

Step-by-Step Solution

Answer

The simplified result of the expression \(10^{2}-100 \div 5^{2} \cdot 2-1\) is 91.

Step 1: Identify and Execute Exponentiation

Two exponentiation operations are present in the expression: \(10^{2}\) and \(5^{2}\). We solve these parts first: \(10^{2} = 100\) and \(5^{2} = 25\). So, the simplified expression becomes \(100 - 100 \div 25 \cdot 2 - 1\).

Step 2: Execute Division and Multiplication

Next, perform the division: \(100 \div 25 = 4\). After that, perform the multiplication: \(4 \cdot 2 = 8\). Thus, the expression simplifies to \(100 - 8 - 1\).

Step 3: Execute Subtraction

Finally, perform the subtraction operations: \(100 - 8 = 92\) and then \(92 - 1 = 91\).

Step 4: Final Result Enlisting

After performing all the operations in the correct order, the original expression \(10^{2}-100 \div 5^{2} \cdot 2-1\) simplifies to 91.

Key Concepts

Simplifying Algebraic Expressions

Simplifying algebraic expressions is a fundamental skill in algebra that involves reducing complex expressions down to a simpler form without changing their value. This simplification process often requires combining like terms, factoring, expanding, and reducing fractions among other techniques.

Let's consider an algebraic expression as an example: \(10^{2}-100 \div 5^{2} \cdot 2-1\). Simplification starts with executing operations in the correct order, which can be remembered using the PEMDAS/BODMAS rules. The goal is to decrease potential errors and ensure that anyone who computes the expression arrives at the same value.

Simplification is not just about making an expression shorter; it often makes the numbers and operations more manageable. For example, performing the exponentiation steps first transforms \(10^{2}\) into 100 and \(5^{2}\) into 25, making further calculations more straightforward.

Exponentiation in Algebra

Exponentiation is an arithmetic operation that involves raising a number, known as the base, to the power of an exponent. In algebra, understanding exponentiation is crucial because it ranks immediately after parentheses in the order of operations hierarchy.

In the expression \(10^{2}-100 \div 5^{2} \cdot 2-1\), the exponentiation operations are \(10^{2}\), where 10 is the base and 2 is the exponent, and \(5^{2}\), where 5 is the base and again 2 is the exponent. Here's what happens during exponentiation: \(10^{2} = 10 \times 10 = 100\) and \(5^{2} = 5 \times 5 = 25\).

This step is vital; if misunderstood or computed incorrectly, it can lead to a completely different outcome. It's important not only to perform exponentiation correctly but to do it before engaging in multiplication, division, addition, or subtraction, as detailed in the order of operations rules.

PEMDAS/BODMAS Rules

The PEMDAS/BODMAS rules provide a clear structure for performing arithmetic operations in the correct order. The acronyms stand for Parentheses/Brackets, Exponents/Orders (indices), Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

In the given exercise, \(10^{2}-100 \div 5^{2} \cdot 2-1\), PEMDAS/BODMAS instructs us to first address any operations within parentheses or brackets, followed by exponents. Once exponentiation is done, we proceed with multiplication and division (working from left to right as they occur), and finally, perform any addition or subtraction.

Following these rules, we first resolve the exponentiation: \(10^{2}\) turns into 100, and \(5^{2}\) simplifies to 25. We then divide and multiply: \(100 \div 25 = 4\) and \(4 \cdot 2 = 8\). Lastly, we execute the subtraction steps, simplifying the expression to 91. Correctly applying this order is essential for achieving the right answer in algebraic problems.