Problem 76 - Algebra A Combined Function

Question

Perform the indicated operation. \(-8(-11)\)

Step-by-Step Solution

Answer

The product of \(-8\) and \(-11\) is 88.

Step 1: Identify the Operation

The given expression is \(-8(-11)\). This indicates a multiplication operation between the two numbers. Multiplying a negative number by another negative number results in a positive product.

Step 2: Multiply the Numbers

To perform the multiplication, multiply the absolute values of the numbers. The absolute value of \(-8\) is \(8\), and the absolute value of \(-11\) is \(11\). Multiply these values:\[8 \times 11 = 88\]

Step 3: Determine the Sign of the Product

Since both numbers in the expression are negative, the product of two negative numbers is positive, according to the rule of multiplication for negative numbers. Therefore, the result is positive \(88\).

Key Concepts

Negative Numbers

Negative numbers might seem tricky, but they follow simple rules. A negative number is any number less than zero. It is usually represented with a minus sign (-). For example, -8 is a negative number. Negative numbers are used to represent concepts like debt, temperatures below zero, or floors below ground level.

When you multiply two negative numbers together, the product is always positive. This might seem strange at first. But think of two negatives as a way of reversing direction twice in a row. Reversing twice takes you back to the start, which is why the product ends up being positive.

• Example: \(-5(-3) = 15\)

Mathematically, the rule is:

• If an even number of negative numbers are multiplied, the product is positive.
• If an odd number of negative numbers are multiplied, the product is negative.

Absolute Value

The absolute value of a number tells you how far away it is from zero on a number line, without worrying about direction. Think of it as the "size" or "magnitude" of a number.

It's denoted by vertical bars |x|. For example, \(|-8|\) equals 8, and \(|11|\) equals 11.

• Absolute value makes calculations easier by removing the negative or positive signs temporarily.

When you multiply numbers, you can work with their absolute values and apply the sign rules separately.

In the expression \(-8(-11)\), both numbers are negative. We first multiply their absolute values: \[8 \times 11 = 88\] and then apply the rule that the product of two negative numbers is positive, giving us positive 88.

Positive Product

A positive product results when two negative numbers or two positive numbers are multiplied. The concept might feel counterintuitive with negative numbers because negatives often suggest "less" or "opposite." But when multiplied, two negatives make a positive.

Here's why: imagine each negative as a reversal. Multiply by -1 reverses direction once. Multiply by -1 again flips it back to the original positive space.

• Multiplying \(-2(-4)\) will yield a positive 8, because a double reversal nullifies the negative effect.

The rule of signs in multiplication is straightforward:

• Negative × Negative = Positive
• Positive × Positive = Positive
• Positive × Negative = Negative
• Negative × Positive = Negative

This understanding simplifies and balances all expressions involving multiplication of integers.