Problem 9 - College Algebra Essentials

Question

Evaluate each expression in Exercises \(1-12,\) or indicate that the root is not a real number. $$\sqrt{25}-\sqrt{16}$$

Step-by-Step Solution

Answer

The result is 1

Step 1: Calculate the square root of the first number

The square root of 25 is 5, because \(5 * 5 = 25\).

Step 2: Calculate the square root of the second number

Similarly, the square root of 16 is 4, because \(4 * 4 = 16\).

Step 3: Subtraction

Subtract the square root of the second number from that of the first number. This gives \(5 - 4 = 1\).

Key Concepts

Real Numbers

Real numbers are the bedrock of basic algebra and arithmetic operations. They encompass all the numbers that one would typically encounter, including integers (like -2, 0, 1), whole numbers, fractions, and irrational numbers like square roots. When we talk about square roots, as seen in the exercise, we are often dealing with real numbers. For instance, both 5 and 4 are real numbers, and these are the square roots of 25 and 16, respectively.

• Real numbers can be positive, negative, or zero.
• They are used in many areas, such as measuring, counting, or performing calculations.
• All operations involving real numbers will yield another real number, as long as we are not taking the square root of a negative number.

Understanding real numbers helps in organizing numbers in an orderly fashion, which is crucial when performing arithmetic operations.

Arithmetic Operations

Arithmetic operations are the fundamental actions we perform on numbers. The main operations are addition, subtraction, multiplication, and division. In the context of square roots, these operations often focus on simplifying expressions to find a numerical solution.
For the given exercise, starting with subtraction, after determining each square root, illustrates a basic arithmetic operation:

• Finding Square Roots: This involves determining a number that, when multiplied by itself, gives the original number. For example, both 25 and 16 have easy-to-calculate square roots which are 5 and 4 respectively.
• Subtraction: After finding the square roots, we subtract 4 from 5, completing the arithmetic operation, resulting in the difference of 1.

These operations reinforce the importance of precise calculation, especially during more complex algebraic expressions.

Basic Algebra

Basic algebra involves using numbers and symbols to make calculations and represent mathematical relationships. It lays the foundation for solving equations and understanding mathematical concepts crucial for higher learning. The exercise to evaluate \(\sqrt{25}-\sqrt{16}\) is an example of a simple algebraic expression involving square roots.
In this context:

• Evaluating Expressions: This step involves performing operations in the correct order, ensuring each component of the expression is simplified properly. Simplicity is achieved by reducing square roots and then addressing arithmetic operations.
• Symbol Manipulation: Algebra often requires rearrangement of symbols and values, indicating operations to be performed, as is the case in combining numbers after evaluating square roots.

Algebra teaches one to think critically about how to rearrange and simplify mathematical expressions, providing the skills needed to solve more complex mathematical problems effectively. By practicing these fundamentals, learners build a strong base for tackling more advanced topics.