Problem 10 - Fundamentals of Mathematics

Question

Find the value of each expression. $$3[-40-2(4 a-3 b)], \text { if } a=-6 \text { and } b=0$$

Step-by-Step Solution

Answer

The value of the expression is 24.

Step 1: Substitute the values of 'a' and 'b' into the expression

First, replace the variables 'a' and 'b' in the expression with their given values: \(-6\) for \(a\) and \(0\) for \(b\). This changes the expression to: \[3 \times [-40 - 2(4(-6) - 3(0))]\].

Step 2: Simplify inside the parentheses

Start with calculating the innermost operations: 1. Compute \(4(-6)\) which is \(-24\).2. Compute \(-3(0)\) which is \(0\). Replace the calculations back into the expression, resulting in: \[3 \times [-40 - 2(-24 - 0)]\].

Step 3: Calculate inside the brackets

Continue simplifying inside the brackets by computing:\(-24 - 0 = -24\).Now compute the expression further: \[3 \times [-40 - 2(-24)]\].

Step 4: Perform multiplication within brackets

Multiply \(-2\) by \(-24\) to get:\(48\).Now, substitute this back into the expression: \[3 \times [-40 + 48]\].

Step 5: Simplify inside the brackets

Perform the arithmetic inside the brackets: \(-40 + 48\ = 8\).So, the expression reduces to \(3 \times 8\).

Step 6: Calculate the final multiplication

Multiply the remaining terms: \(3 \times 8 = 24\).

Key Concepts

Simplifying Expressions

Simplifying expressions in algebra is all about making complex mathematical phrases shorter or easier to understand. It involves reducing an expression to its simplest form without changing its value. By simplifying, you aim to break down any complicated parts, making them easier to work with and understand.

**Key Steps to Simplify:**

• **Identify like terms**: Combine terms that have the same variable raised to the same power.
• **Use the distributive property**: This helps in expanding expressions like \(a(b + c)\) into \(ab + ac\).
• **Remove parentheses**: Once values are substituted in, you start by simplifying what's inside the parentheses first, to avoid errors.

In our example, simplifying involved substituting the values of the variables and then carrying out operations inside the parentheses and brackets step by step. This systematic approach ensures accuracy and consistency in simplifying the original expression.

Order of Operations

Order of operations is a fundamental principle in mathematics that dictates the correct order to evaluate a mathematical expression. By following these rules, you ensure that the answer you get is accurate. To remember the order, you might use the acronym PEMDAS:

• **P** for Parentheses
• **E** for Exponents
• **M/D** for Multiplication and Division (from left to right)
• **A/S** for Addition and Subtraction (from left to right)

In our problem, following the order of operations was crucial. We started by evaluating what was inside the parentheses, and then moved step by step through multiplication and subtraction as necessary. This method prevents mistakes that could arise from computing terms in the wrong order, leading to incorrect results.

Arithmetic with Integers

Arithmetic with integers involves performing basic mathematical operations like addition, subtraction, multiplication, and division with whole numbers, including negative numbers. Understanding this is crucial for algebra, as integer operations form a foundation for more complex calculations.

**Operations with Integers:**

• **Addition and Subtraction**: Use the number line to visualize operations. Moving right indicates addition, while moving left indicates subtraction.
• **Multiplication and Division**: Remember the rules for signs:
  • Positive × Positive = Positive
  • Negative × Negative = Positive
  • Positive × Negative = Negative
  • Negative × Positive = Negative

In our exercise, integer arithmetic was used when we performed operations like multiplying \(-2\) by \(-24\) to get \(+48\), and adding negative and positive numbers within the brackets. These basic integer rules guide us in achieving correct solutions consistently.