Problem 132 - Beginning and Intermediate Algebra

Question

Write each of the following as an expression and evaluate. The sum of 1 and the product of \(-8\) and \(-5\)

Step-by-Step Solution

Answer

The result is 41.

Step 1: Understand the terms and operations

First, let's break down the given statement: 'The sum of 1 and the product of -8 and -5'. This means we need to first find the product of the numbers -8 and -5, and then add this product to 1.

Step 2: Find the product of -8 and -5

The product of any two numbers is found by multiplying them together. Therefore, we need to multiply -8 by -5: \((-8) \times (-5)\)When multiplying two negative numbers, the product is positive. So the result is:\(40\)

Step 3: Add the product to 1

Now that we have the product, 40, we add this to 1 to find the sum: \(1 + 40\)This results in:\(41\)

Step 4: Final answer

The final expression, combining both operations, is \(1 + ((-8) \times (-5))\) After calculation, the value of this expression is 41.

Key Concepts

Integer Operations

Integer operations include actions like addition, subtraction, multiplication, and division involving whole numbers. Integers are numbers without fractions or decimals and can be positive, negative, or zero.
To work effectively with integers, you must understand their rules:

• Addition of two positive integers or two negative integers gives a sum that retains the sign of the integers involved.
• When you add a negative integer and a positive integer, you essentially subtract the smaller value from the larger one and take the sign of the larger value.
• Subtraction is the same as adding the additive inverse. For example, subtracting a number is the same as adding its negative.
• Multiplying two integers follows a sign rule: the product of two integers with the same sign is positive, and with different signs, it is negative.

Understanding these basic operations is crucial to solving algebraic problems, like evaluating expressions with integers. Notice how crucial these operations are when you handle mathematical expressions in your coursework.

Order of Operations

The order of operations is a set of rules used to clarify which procedures should be performed first when evaluating a mathematical expression. It ensures that everyone can interpret and solve expressions consistently.

• Start with any operations inside parentheses or brackets.
• Then, perform exponentiation (powers and roots), if any exist.
• Multiplication and division are next, moving from left to right.
• Lastly, tackle addition and subtraction, also going from left to right.

Without this order, calculations can yield different and incorrect results. Remember the mnemonic "PEMDAS" (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) to recall this sequence. In our example, we dealt with multiplication before moving on to addition, adhering to the rules of PEMDAS. That's why knowing the order is vital, as it helps break down complex expressions step-by-step.

Multiplication of Integers

Multiplication of integers is a fundamental operation in mathematics that involves combining equal groupings of a number. When multiplying integers, remember that the product's sign depends on the signs of the integers being multiplied.

• If both integers have the same sign, the product is positive.
• If the integers have different signs, the product is negative.

In our exercise, we multiply \((-8)\) and \((-5)\). Both numbers are negative. According to the sign rule, the product is positive: \((-8) \times (-5) = 40\). Understanding these rules simplifies solving algebraic expressions that require multiplying several integers, ensuring you arrive at the correct solution.