Problem 35 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. $$ (16 \cdot 25)^{1 / 2} $$

Step-by-Step Solution

Answer

The answer to the expression \((16 \cdot 25)^{1 / 2}\) is 20.

Step 1: Simplify inside the Brackets

According to BODMAS, calculations in brackets are done first. Multiply the numbers inside the brackets. This gives 16 \times 25 = 400.

Step 2: Evaluate the Square Root

The square root of 400 is evaluated, which is the principal square root. The square root of 400 is 20.

Key Concepts

BODMAS Rule

When evaluating mathematical expressions, it's essential to follow the BODMAS rule. This ensures operations are performed in the correct order, which is critical for getting the right result. BODMAS stands for:

• Brackets first
• Orders (i.e., powers and roots, etc.)
• Division and Multiplication (from left to right)
• Addition and Subtraction (from left to right)

By sticking to these guidelines, complex expressions can be simplified properlly. In the given exercise, we first address the operation within the brackets, thus following the BODMAS rule. This ensures no steps are overlooked.

Simplifying Expressions

Simplifying expressions involves breaking down complex calculations into simpler parts. The aim is to make the expression easier to work with.

In the example provided, we begin by simplifying the expression inside the brackets by multiplying the numbers 16 and 25. This process eliminates the brackets and reduces the expression to a single multiplication task:

• First, calculate the multiplication: \(16 \times 25 = 400\).

With the multiplication complete, the expression becomes \(400^{1/2}\). Simplifying it any further, such as evaluating the square root, is the next step. Always ensure each operation is completed before moving on to the next task in the expression.

Principal Square Root

The principal square root refers to the positive square root of a number. When evaluating expressions with square roots, such as \(400^{1/2}\), it’s important to find this primary positive value.

The square root operation, \( \sqrt{400} \), involves determining what number, when multiplied by itself, gives 400. In this case, the principal square root of 400 is 20 because:

• \(20 \times 20 = 400\)

Understanding that the principal square root is always positive helps simplify expressions correctly. It ensures calculations adhere to conventional mathematical rules and interpretations.