Problem 42 - Fundamentals of Mathematics

Question

Use the order of operations to determine each value. \(\sqrt{18 \cdot 2}\)

Step-by-Step Solution

Answer

The value is 6.

Step 1: Clarify Inner Expression

The expression inside the square root is \(18 \cdot 2\). We will need to calculate this product first as part of simplifying inside the root.

Step 2: Calculate the Product

Multiply the two numbers: \(18 \times 2 = 36\). So the expression inside the square root now becomes \(\sqrt{36}\).

Step 3: Simplify the Square Root

Find the square root of the result from Step 2. \(\sqrt{36} = 6\) because 6 multiplied by 6 gives 36.

Key Concepts

Square Root

The square root is a fascinating mathematical concept that basically asks, "What number multiplied by itself gives this product?" In the expression \( \sqrt{36} \), we need to find a number which when squared (i.e., multiplied by itself), results in 36.
One important point to remember is that there are often two numbers that can serve as square roots for a given number – one positive and one negative. For example:

• The square root of 36 is 6 because \( 6 \times 6 = 36 \).
• But, -6 is also a square root because \( (-6) \times (-6) = 36 \).

However, in most mathematical problems, especially in basic arithmetic operations, we usually consider only the principal (positive) square root unless specified otherwise.

Multiplication

Multiplication is one of the basic arithmetic operations where you calculate the total of one number added to itself a certain number of times. In the expression \(18 \times 2\), you are essentially adding 18 two times or 2 eighteen times, both leading to the product 36.
The operation can be simplified by using repeated addition or utilizing memorized multiplication facts:

• For instance, \(18 \times 2 = 36\) because adding 18 to itself gives 36.
• This operation is commutative, meaning \(18 \times 2\) yields the same result as \(2 \times 18\).

Multiplication is an essential step in performing calculations involving both numbers and variables, and it forms the backbone of solving more complex expressions efficiently.

Simplification

Simplification is the process of reducing a math expression to its simplest form, making it easier to solve or understand. The simplification often involves applying the order of operations, managing expressions within parentheses, evaluating exponents, and executing basic arithmetic operations.
In our exercise, simplifying the expression inside the square root is pivotal. Here's a streamlined approach:

• First, perform the multiplication: \(18 \times 2 = 36\).
• Then, calculate the square root: \(\sqrt{36} = 6\).

By breaking the problem down into smaller, manageable steps, you ensure clarity and accuracy in obtaining your final answer. This logical approach can be applied to any mathematical problem, ensuring you don't miss any critical steps in your calculations.