Problem 14 - Prealgebra

Question

Simplify each of the following expressions without using a calculator. $$8 \sqrt{36}$$

Step-by-Step Solution

Answer

The simplified expression is 48.

Step 1: Identify the square root term

In the expression, identify the term that involves a square root. Here, it is \( \sqrt{36} \).

Step 2: Simplify the square root

Find the square root of 36. Since 36 is a perfect square, \( \sqrt{36} = 6 \).

Step 3: Multiply with the scalar

Multiply the simplified square root result by the scalar factor outside the square root. In this case, multiply 8 by 6, which gives: \( 8 \times 6 = 48 \).

Step 4: Finalize the simplified expression

The simplified expression of \( 8 \sqrt{36} \) is \( 48 \).

Key Concepts

Understanding Prealgebra Problems

Prealgebra serves as the foundation for all future math courses. At this stage, students start to encounter problems that require an understanding of fundamental arithmetic operations and basic algebraic principles. Breaking down expressions and simplifying them, as we did with the original problem \[8 \sqrt{36}\], helps students build confidence in handling more complex equations. Simplifying expressions involves identifying what each part of the expression represents and how they relate to each other. This skill will be essential as you progress into higher levels of math, such as algebra and beyond. As you work through prealgebra, you'll often find that following a step-by-step approach, much like the solution provided, will make these expressions easier to manage. Evaluating parts of an equation separately before combining them, as demonstrated in the solution, is a common strategy used in math.

Mastering Basic Arithmetic Operations

Understanding how to perform basic arithmetic operations is crucial not only for simplifying square roots but also for solving a wide range of math problems. These operations include addition, subtraction, multiplication, and division. * Addition adds two numbers together. * Subtraction removes one number from another. * Multiplication is repeated addition. * Division splits a number into equal parts. In our exercise, multiplication played a key role. We started with simplifying the square root of 36 to get 6. Next, the multiplication of 8 by 6 gives us 48. Mastering these basic operations allows us to transform complex problems into simpler ones.

Perfect Squares and Their Importance

Perfect squares are numbers that result from multiplying an integer by itself. Recognizing them can simplify roots significantly. For instance, numbers such as 4, 9, 16, 25, and indeed 36, are perfect squares. This means their square roots are whole numbers. Understanding this concept speeds up the simplification process in problems like\[8 \sqrt{36}\], since it allows you to quickly identify that\[\sqrt{36} = 6.\]Knowing which numbers are perfect squares is extremely helpful when simplifying square roots because it eliminates any guesswork about the value of the root.