Problem 15 - Precalculus

Question

Evaluate each expression. $$|7(-8)|-|7| \cdot|-8|$$

Step-by-Step Solution

Answer

The value of the expression is 0.

Step 1: Evaluate Inside the Absolute Value

Begin by solving the expression inside the first absolute value: \(7(-8)\). This simplifies to \(-56\).

Step 2: Apply Absolute Value

Evaluate the absolute value of \(-56\), which is \(|-56| = 56\).

Step 3: Evaluate Absolute Value for Each Term Individually

Find the absolute value of \(7\) and \(-8\) individually. \(|7| = 7\) and \(|-8| = 8\).

Step 4: Multiply the Absolute Values

Multiply the results of the absolute value from Step 3: \(7 \times 8 = 56\).

Step 5: Subtract the Products

Subtract the second result from the first result: \(56 - 56 = 0\).

Key Concepts

Understanding Multiplication

Multiplication is a fundamental arithmetic operation that involves combining groups of equal size. In arithmetic, when we multiply two numbers, we are essentially adding a number to itself multiple times. For example, multiplying 7 by -8, written as \(7 \times -8\), is the same as adding -8 to itself seven times. This process results in -56.

• When multiplying, if both numbers have the same sign (both positive or both negative), the result is positive.
• If the numbers have opposite signs (one positive and one negative), the result becomes negative.

In the given exercise, we see this operation performed where \(7 \times -8 = -56\). Understanding the effects of the signs in multiplication is critical for correctly evaluating expressions involving both positive and negative numbers.

The Process of Subtraction

Subtraction is the process of finding the difference between numbers. It is one of the basic arithmetic operations where you take away a certain amount from a larger amount. Subtraction is the opposite of addition.

• When you subtract two numbers, you determine how much one number is less than the other.
• In expressions, subtraction is often denoted by the minus symbol \(-\).

In the given expression \(56 - 56\), subtraction is used to find the difference between two equal values. When you subtract a number from itself, such as 56 from 56, the difference is always zero, reflecting an important property of subtraction.

Evaluating Expressions

Evaluating expressions involves simplifying an expression to find its value. This often requires following specific steps to handle operations like addition, subtraction, multiplication, and division, often dealing with the order of operations.

• Operations within parentheses or absolute values are calculated first.
• Multiplication and division are performed before addition and subtraction.

In the expression \(|7(-8)| - |7| \cdot |-8|\), evaluating expressions means dealing with absolute values, multiplying and then subtracting. First, you evaluate inside the absolute values, convert any negative results into positive by taking the absolute value, and substitute these into the expression.
Understanding how to evaluate expressions is crucial for accurately finding the result, as shown where we calculated the expression to finally result in zero after following all these steps.