Problem 67 - Elementary Algebra

Question

For exercises 1-80, evaluate. $$ \frac{(9-1)^{2}-64}{2^{3}} $$

Step-by-Step Solution

Answer

0

Step 1 - Simplify the expression inside the parentheses

Evaluate the expression inside the parentheses first: The expression is (9-1) = 8

Step 2 - Calculate the exponent

Now, raise the simplified result to the power indicated: (8)^2 = 64

Step 3 - Subtract 64 from the result

Subtract 64 from the squared value: 64 - 64 = 0

Step 4 - Evaluate the denominator

Evaluate the denominator: 2^3 = 8

Step 5 - Divide the numerator by the denominator

Finally, divide the result of the numerator by the result of the denominator: 0 / 8 = 0

Key Concepts

Simplifying Expressions

Simplifying expressions means breaking down complex mathematical expressions into simpler forms. This helps in making calculations easier and faster. In our example, the first step is to evaluate the expression inside the parentheses. We have \(9 - 1\), which simplifies to \(8\). When simplifying expressions:

• Always work within parentheses first.
• Simplify any operations inside the parentheses before moving on.

By breaking down the problem into smaller parts, we make it manageable and easy to understand. Remember, the goal is to make the expression less complex at each step.

Order of Operations

The order of operations is crucial when solving mathematical expressions. It’s often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)). For the given problem, we need to follow this order:

• First, solve the expression within the parentheses: \(9 - 1 = 8\).
• Second, handle the exponents: \(8^2 = 64\).
• Next, continue with any subtraction: \(64 - 64 = 0\).
• Then, evaluate the denominator: \(2^3 = 8\).
• Finally, perform the division: \frac{0}{8} = 0\.

Following the order ensures that we arrive at the correct solution every time. Skipping or mixing the order can lead to mistakes.

Basic Algebra

Basic algebra is the branch of mathematics that deals with variables and the rules for manipulating these variables. It includes operations such as addition, subtraction, multiplication, and division. In our example, although we aren't dealing directly with variables, we're using fundamental algebraic principles:

• Simplifying expressions within parentheses.
• Applying exponents to numbers.
• Performing arithmetic operations.

Understanding these basic concepts is crucial as they form the foundation for more advanced topics in algebra and other areas of mathematics. Always remember that each algebraic problem can be broken down into small, manageable steps.