Problem 1 - Algebra for College Students

Question

For Problems \(1-20\), evaluate each of the following. For example, \(\sqrt{25}=5\). \(\sqrt{64}\)

Step-by-Step Solution

Answer

The square root of 64 is 8.

Step 1: Understanding the Problem

Identify the problem as finding the square root of a given number, which in this case is 64.

Step 2: Recalling the Definition of Square Root

Recall that the square root of a number is a value that, when multiplied by itself, gives the original number. So, we are looking for a number which, when squared, equals 64.

Step 3: Finding the Square Root

Recognize that 8 is the number that multiplied by itself results in 64 since \(8 \times 8 = 64\). Thus, \(\sqrt{64} = 8\).

Key Concepts

Algebra Problems

Algebra problems often involve finding unknown values based on rules and relationships between numbers. These problems can range from simple calculations to more complex equations. The example given in the exercise, finding the square root of 64, is a fundamental algebraic concept. Solving this involves understanding what a square root is and recognizing numerical patterns.
Let's break it down further: when tasked with such problems, work on reorganizing the problem statement, identify which mathematical operation to employ, and then apply the relevant algebraic step-by-step approach.
This type of problem lays the groundwork for more advanced algebraic equations involving variables and unknowns.

Number Operations

Mathematical number operations involve the basic procedures you perform on numbers: addition, subtraction, multiplication, division, and more complex operations like finding roots or powers. In our specific task—finding the square root of 64—it's part of a broader family of operations aimed at simplifying or solving expressions.
Here's how these operations are interconnected: finding a square root is essentially the inverse of calculating a square (a number multiplied by itself). Number operations like these are crucial when working through algebra problems since they form the building blocks for more complex calculations.
By mastering these basic operations, you can seamlessly transition to dealing with equations that incorporate variables and require manipulation using multiple methods.

Mathematical Definitions

Understanding mathematical definitions is fundamental in solving algebra problems, such as determining the square root. The square root of a number is defined as a number that, when multiplied by itself, equals the original number. It is often denoted using the radical symbol \( \sqrt{} \).
In our example, the square root of 64 is the number that satisfies \( x \times x = 64 \), which is 8. This definition not only helps in solving for square roots but is also applicable in various mathematical contexts, making it a versatile tool in problem-solving.

• This concept allows mathematicians to reverse the squaring process.
• Understanding these definitions aids in recognizing patterns within numbers.
• They serve as the backbone for learning more about functions and equations in mathematics.

Emphasizing definitions helps solidify your understanding of why and how certain operations are performed.