Problem 45 - Fundamentals of Mathematics

Question

Use the order of operations to determine each value. \(\sqrt{8 \cdot 8}\)

Step-by-Step Solution

Answer

The value is 8.

Step 1 - Identify the Expression Inside the Square Root

The expression to evaluate inside the square root is \(8 \cdot 8\). According to the order of operations (PEMDAS/BODMAS), multiplication should be performed before applying the square root.

Step 2 - Perform Multiplication

Multiply the numbers inside the square root: \(8 \cdot 8 = 64\). Now, the expression simplifies to \(\sqrt{64}\).

Step 3 - Calculate the Square Root

Determine the square root of 64. Since \(8^2 = 64\), the square root of 64 is 8. Therefore, \(\sqrt{64} = 8\).

Key Concepts

Square Root

Understanding square roots is essential in mathematics. A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 64 is 8 because:

• 8 multiplied by 8 is 64

You will often see the square root symbol as \(\sqrt{x}\), where \(x\) is the number under the square root. Calculating square roots helps in simplifying expressions and solving equations. When you see a square root in an expression, it’s often one of the last operations you'll perform, as you first deal with the operations inside the square root symbol.

Multiplication

Multiplication is one of the four basic arithmetic operations. It's a quicker way of adding the same number multiple times. In the expression \(8 \cdot 8 = 64\), you are finding the product of 8 with itself.

• Multiplication here is straightforward: 8 times 8
• The result is 64

Performing multiplication correctly is crucial, as it directly affects the outcome of the final calculation, especially when other operations follow.

PEMDAS

PEMDAS is an acronym used to remember the order of operations in math. It stands for:

• P – Parentheses
• E – Exponents (or powers and roots, like the square root)
• M – Multiplication
• D – Division
• A – Addition
• S – Subtraction

Using PEMDAS ensures that you perform calculations in the correct sequence. For example, when you face an operation like \(\sqrt{8 \cdot 8}\), you first multiply 8 by 8 before taking the square root of the result. This order is crucial to getting the correct answer.

BODMAS

BODMAS is another acronym similar to PEMDAS. It helps remember the sequence of operations:

• B – Brackets
• O – Orders (which include square roots and exponents)
• D – Division
• M – Multiplication
• A – Addition
• S – Subtraction

BODMAS often serves as a quick guide to solving mathematical expressions correctly. Like in PEMDAS, you multiply what's inside the square root before determining the square root itself. Remembering BODMAS or PEMDAS can help keep your math problems on track.