Problem 66 - College Algebra

Question

Determine whether the simplifi d expression is rational or irrational: \(\sqrt{-16+4(5)+5}\).

Step-by-Step Solution

Answer

The expression is rational.

Step 1: Simplify the Expression Inside the Square Root

Start by simplifying the expression inside the square root: \(-16 + 4(5) + 5\).Multiply \(4\) by \(5\), which gives \(20\).Substitute back into the expression: \(-16 + 20 + 5\).

Step 2: Perform Addition and Subtraction

Continue simplifying the expression:- Perform the addition and subtraction in order: - First, calculate \(-16 + 20\), which is equal to \(4\). - Then add \(5\) to \(4\), resulting in \(9\).So, the expression inside the square root simplifies to \(9\).

Step 3: Evaluate the Square Root

Now, take the square root of the simplified expression:\(\sqrt{9} = 3\).Here, \(3\) is a whole number.

Step 4: Determine if the Result is Rational or Irrational

Since \(3\) is a whole number, it is a rational number (rational numbers can be expressed as a fraction of two integers).

Key Concepts

Simplifying Expressions

To simplify an expression, you need to reduce it to its simplest form. This often involves performing operations such as multiplication, addition, and subtraction in a specific order, known as the order of operations. Here is a quick guide to simplify expressions effectively:

1. **Examine and simplify operations within parentheses first**: Just like our problem, the expression inside the square root was simplified step by step. Inside the expression \(-16 + 4(5) + 5\), multiplication takes precedence, which means we start by calculating \(4 \times 5\).
2. **Perform additions and subtractions next**: Once the multiplication is resolved, you can move on to the next step, adding \(20\) to the rest of the terms: \(-16 + 20 + 5\). Evaluate these one by one: \(-16 + 20 = 4\) and then \(4 + 5 = 9\).
3. **Combine steps efficiently**: Always make sure each step logically follows the others. When the remaining expression is as simplified as possible, you've successfully simplified the problem.

Simplifying expressions is crucial to understanding more complex math problems and ensuring correct calculations. Practice will deepen your confidence and proficiency in handling these types of tasks.

Square Root

The square root is a mathematical concept that involves finding the number which, when multiplied by itself, produces the given number. In our example, once the expression inside the square root is simplified to \(9\), we take its square root.

Let's dive into the steps for evaluating the square root effectively:
- **Identify the perfect squares**: A perfect square is a number that is the square of an integer. In our problem, \(9\) is a perfect square since \(3 \times 3 = 9\).
- **Apply the square root operation**: The symbol \(\sqrt{}\) denotes the principal square root, which means the non-negative root of a number. Thus, \(\sqrt{9} = 3\).

Understanding square roots is an essential skill for solving mathematical equations involving powers and roots. Being able to identify perfect squares quickly will make the process faster, and the solutions clearer.

Integer Operations

Integer operations involve basic arithmetic actions such as addition, subtraction, multiplication, and division for whole numbers. In our solved exercise, integer operations are crucial to simplifying the expression before taking the square root.

Here's how you can handle integer operations effectively:
- **Add and subtract carefully**: In our original expression, once the multiplication is handled, perform any addition or subtraction remaining. Combining integers correctly can make or break your solution:
- For \(-16 + 20\), add \(20\) to \(-16\) to get \(4\).
- Then, add \(4 + 5 = 9\).

In conclusion, mastering the basics of integer operations will provide clarity and accuracy in dealing with more complex equations. These operations form the cornerstone of algebra and many other math concepts, providing the foundation to tackle wider mathematical challenges.