Question

((7 @(4 \\# 5))\( (? 3): (a) 204 (b) 306 (c) 408 (d) 102 # Find the value of \)2 \\#((7 @(4 \\# 5))$ (? 3): (a) 204 (b) 306 (c) 408 (d) 102

Short answer

3) if \(\\#\) represents multiplication, \(@\) represents addition, and \(?\) represents subtraction? Answer: 48

Step-by-step solution

Calculate inside parentheses

Follow the order of operations (PEMDAS) to calculate the values inside the parentheses. First, calculate the multiplication and then the addition. \(2 \times ((7 +(4 \times 5)) - 3)\)

Calculate multiplication inside parentheses

Calculate the value of \((4 \times 5)\): \(4 \times 5 = 20\) Now, replace the \((4 \times 5)\) with 20: \(2 \times ((7 + 20) - 3)\)

Calculate addition inside parentheses

Calculate the value of \((7 + 20)\): \(7 + 20 = 27\) Now, replace the \((7 + 20)\) with 27: \(2 \times (27 - 3)\)

Calculate subtraction

Calculate the value of \((27 - 3)\): \(27 - 3 = 24\) Now, replace the \((27 - 3)\) with 24: \(2 \times 24\)

Final calculation

Calculate the value of \((2 \times 24)\): \(2 \times 24 = 48\) So, the final value of the given expression is 48 (which is not one of the available answer choices).

Key concepts

PEMDAS

When solving mathematical expressions, understanding the order of operations is crucial. This is commonly remembered by the acronym **PEMDAS**, which stands for **Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction**. These steps serve as a guideline to solve expressions systematically, ensuring you retrieve the correct result.

• **Parentheses** are addressed first, which makes any calculations within them the utmost priority.
• **Exponents** come next, though this does not apply to our current expression.
• Follow this with **Multiplication** and **Division**, which are processed from left to right.
• The final steps are **Addition** and **Subtraction**, also performed from left to right.

In our example, the expression was approached step-by-step using PEMDAS, starting with operations inside the parentheses. First, the arithmetic operation of multiplication was applied, followed by addition inside the parentheses, prior to any subtraction or multiplication outside of it.

arithmetic operations

Arithmetic operations are the basic mathematical practices used to perform calculations. They form the core of algebra and involve four primary actions: addition, subtraction, multiplication, and division.

• **Addition (+)** brings two numbers together to form a total or sum.
• **Subtraction (−)** takes one number away from another, calculating the difference.
• **Multiplication (×)** finds the total of one number added to itself a specified number of times.
• **Division (÷)** splits a number into equal parts.

In the given exercise, these operations were utilized systematically as follows: Multiplication was first applied within the parentheses to determine the product of 4 and 5. Then, addition was used to combine 7 and 20. Lastly, subtraction refined the result further by subtracting 3, following which multiplication finalized the overall value.

expression simplification

Expression simplification is the process of condensing a mathematical expression to reveal its simplest form, making it easier to understand and solve.

Steps you can take to simplify expressions include:

• **Identify and perform arithmetic operations**: Initially manage calculations nested in parentheses, addressing any grouped terms.
• **Combine like terms**: Add or subtract identical variables to simplify further.
• **Apply constant arithmetic rules**: Use known mathematical principles such as the distributive property to reduce complex expressions.

In our expression, simplification involved multiple calculations within and outside the parentheses. Each step simplified the problem incrementally: combining results until only a straightforward operation was left, ensuring clarity and reducing any chance for mistakes. Always approach such expressions methodically to achieve the correct outcome effortlessly.