Problem 62 - Algebra A Combined Function

Question

Perform the indicated operation. \(-1.3(-0.6)\)

Step-by-Step Solution

Answer

The product is +0.78.

Step 1: Identify the Operation

The given expression is \(-1.3 \times (-0.6)\). This is a multiplication operation between two negative numbers.

Step 2: Multiply the Absolute Values

First, ignore the negative signs and multiply the absolute values of the two numbers: \(1.3 \times 0.6\). To do this, multiply:\(1.3\) can be written as \(13 \div 10\), and \(0.6\) as \(6 \div 10\);\(13 \times 6 = 78\). Divide this product by 100 to handle the decimal places: \(78 \div 100 = 0.78\).

Step 3: Determine the Sign of the Product

The rule of multiplication dictates that multiplying two negative numbers results in a positive product. Since both numbers in this operation are negative, the product will be positive.

Step 4: Finalize the Answer

Combine the result from Step 2 with the sign determined in Step 3. The product of \(-1.3\) and \(-0.6\) is \(+0.78\).

Key Concepts

Understanding Absolute Values

The absolute value of a number is its distance from zero on the number line, without considering direction or sign. It is always a positive number, or zero itself. Absolute values help us simplify complex calculations, especially when dealing with negative numbers.

• For any number \( x \), the absolute value is denoted as \( |x| \).
• Example: \( |-5| = 5 \) and \( |3| = 3 \).

In the context of multiplication, finding the absolute values means temporarily removing negative signs to evaluate the operation more easily. Once absolutes are known, the multiplication becomes straightforward. The step-by-step solution involved this when it instructed to multiply \( 1.3 \) and \( 0.6 \) instead of considering their signs initially.

Decimal Multiplication Simplified

Decimal numbers can appear tricky at first, but multiplying them follows familiar rules. The challenge lies in correctly placing the decimal point in the final product. Let's break it down:

• First, ignore the decimal points and multiply the numbers as if they were whole numbers.
• Next, count the total number of decimal places from both original numbers.
• Finally, place the decimal in the product, ensuring it reflects the combined decimal places.

For example, when multiplying \( 1.3 \) (one decimal place) by \( 0.6 \) (one decimal place), calculate \( 13 \times 6 = 78 \). There are two decimal places in total, so position the decimal to give \( 0.78 \). This reflects that both numbers were tenths.

Positive Product from Negative Numbers

Multiplying negative numbers may seem perplexing at first, but it becomes logical with the right understanding. When you multiply two negative numbers, you end up with a positive product.

• Think of it as two negatives canceling each other out.
• In mathematics, this rule is consistent: the product of two negatives is positive.
• Remember, if you multiply with different signs, the product is negative (e.g., \( -1 \times 1 = -1 \)).

For the given problem, multiplying \( -1.3 \) and \( -0.6 \), we apply the rule: two negatives yield a positive. Thus the product is \( +0.78 \). Understanding this rule assists in predicting the sign of the result in any operation involving negatives.