Problem 43 - Fundamentals of Mathematics

Question

Use the order of operations to determine each value. \(\sqrt{6 \cdot 6}\)

Step-by-Step Solution

Answer

The value is 6.

Step 1: Understand the Order of Operations

The order of operations is a set of rules that dictate the correct sequence to evaluate a mathematical expression. It is commonly remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication/Division (from left to right), Addition/Subtraction (from left to right). In this exercise, we'll first handle the expression inside the square root, which means multiplication.

Step 2: Perform the Multiplication Inside the Square Root

Inside the square root, we have the expression \(6 \cdot 6\). According to PEMDAS, we perform multiplication before taking the square root. Calculate \(6 \cdot 6 = 36\).

Step 3: Take the Square Root

Now that we've simplified the expression inside the square root to 36, we take the square root of 36. The square root of 36 is 6, since \(6^2 = 36\).

Key Concepts

PEMDAS

PEMDAS is a nifty acronym that helps us remember the sequence in which operations are performed in a mathematical expression. It stands for:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

Following PEMDAS ensures that everyone solves math problems in the same way, leading to consistent answers.
Without PEMDAS, you might get confused about which operation to perform first, which could lead to errors. For instance, if you see an expression like \(2 + 3 \times 4\), PEMDAS tells us to first handle multiplication, so you calculate \((3 \times 4)\), and only then add 2, resulting in 14, not 20.
In the context of square roots, we first deal with anything inside the square root, which often involves operations like multiplication before taking the actual square root value.

Square Root

A square root asks what number, when multiplied by itself, results in a given number. It's a mathematical function that undoes squaring a number. For example, the square root of 36 is 6, because when you multiply 6 by itself (6 \times 6), you get 36.
The symbol for square root is \(\sqrt{}\), and it is one of the more delicate operations, typically done after any calculations inside it are completed.
Knowing that \(\sqrt{6 \times 6}\) simplifies to \(\sqrt{36}\), and not coming directly at this square root by skipping multiplication, helps us maintain accuracy in our calculations. Calculating square roots directly is generally straightforward when you're dealing with perfect squares, numbers like 4, 9, 16, 25, etc., which have whole numbers as roots.

Multiplication

Multiplication is a fundamental mathematical operation that combines groups of numbers. It is one of the primary operations used within PEMDAS.
The expression \(6 \times 6\) inside the square root is a straightforward multiplication. Multiplying 6 by 6 helps us determine the value within the square root, yielding 36 as the result.
Because multiplication comes before square roots in the order of operations, it's crucial to first calculate the product of the numbers. This approach ensures that the expression under the square root is simplified correctly before taking further steps.
This process not only makes evaluating more complex mathematical expressions easier but also reduces errors in handling more significant problems that involve repeated or sequential calculations.