Problem 61 - Elementary Algebra

Question

Perform the operations. $$ -8 \cdot 4(-3) \div 2 $$

Step-by-Step Solution

Answer

The result is 48.

Step 1: Address the Grouping Within Parentheses

To start, note that the only grouping involves \(-3\), so we do not need any changes here. We simply move on to the next step, which is multiplication.

Step 2: Perform the Multiplication

Multiply \(-8\) and \(4\), then multiply the result by \(-3\). So, \(-8 \cdot 4 = -32\), and then \(-32 \cdot (-3) = 96\).

Step 3: Perform the Division

Divide \(96\) by \(2\). Therefore, \(96 \div 2 = 48\).

Key Concepts

Understanding Multiplication

Multiplication is an arithmetic operation where you calculate the total of one number being added to itself a specified number of times. It is the same as repeated addition. For example, if you multiply 3 by 4, you are essentially adding 3 together four times, like 3 + 3 + 3 + 3 = 12.

In algebra, multiplication can involve both positive and negative numbers. A key point to remember is:

• When you multiply two numbers with the same sign, whether both positive or both negative, the product is positive.
• If you multiply two numbers with different signs, the product is negative.

For example, in the provided exercise, \(-8 \cdot 4 = -32\), and then \(-32 \cdot (-3) = 96\). Note how the result changed from negative to positive because \(-32\) and \(-3\) are both negative, so their product is positive.

Multiplication is often indicated with the use of the symbol \(\cdot\) or by writing variables or numbers directly next to each other, aiding in simplification.

Grasping Division

Division is the process of splitting a number into equal parts. It's essentially the reverse of multiplication. If you think of multiplication as repeated addition, consider division as repeated subtraction. For example, dividing 12 by 3 is like subtracting 3 from 12 repeatedly until you reach zero, which gives 4.

In terms of signs, division has rules just like multiplication:

• Dividing two numbers with the same sign gives a positive quotient.
• Dividing two numbers with different signs gives a negative quotient.

The exercise involves dividing \(96\) by \(2\), using the formula \(96 \div 2\) which results in \(48\). Here, both numbers are positive, hence the quotient is also positive.

Understanding division ensures that you can accurately distribute amounts or portions in a problem, especially when moving onto more complex algebraic functions.

The Order of Operations

The order of operations is crucial in mathematics to avoid ambiguity and to ensure that everyone gets the same answer. Often remembered by the acronym PEMDAS/BODMAS:

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

In our example, \(-8 \cdot 4(-3) \div 2\),
the calculation started with multiplication followed by division due to this order.

This order ensures operations are performed in a systematic way, allowing us to achieve the same result as others for the same problem. Keep it in mind to tackle more complex problems efficiently.