Problem 46 - Algebra A Combined Function

Question

Perform the indicated operation. (-7)(-7)

Step-by-Step Solution

Answer

49

Step 1: Identify the Operation

The operation indicated is the multiplication of two negative numbers: \((-7)\) and \((-7)\).

Step 2: Recall Multiplication Rule

Recall the multiplication rule that states: 'The product of two negative numbers is positive.' This is applicable in this scenario since both numbers are negative.

Step 3: Multiply the Numbers

Now perform the multiplication of the absolute values of the numbers. Multiply 7 by 7: \(7 \times 7 = 49\).

Step 4: Determine the Sign

According to the multiplication rule, since both numbers were negative, the result of their multiplication is positive. Therefore, the final result is positive 49.

Key Concepts

Negative numbers

When we deal with negative numbers in mathematics, it's crucial to understand that they represent values less than zero. These numbers are located to the left of zero on the number line.
Negative numbers often arise in real-world contexts such as temperatures below freezing or debts in financial situations.
Here are some key points to remember:

• Negative numbers are denoted with a minus sign (-) before them. An example is (-7).
• When we see the same negative number twice, like (-7) and (-7), it indicates we are looking at the same negative value multiple times.
• Negative numbers can be involved in various mathematical operations, such as addition, subtraction, multiplication, and division. Each operation has its own set of rules when negative numbers are involved.

Understanding these basics helps set the foundation for more advanced mathematical operations such as multiplication.

Multiplication rules

When multiplying integers, especially negative numbers, it’s important to know some key rules that simplify the process.
These rules help determine the sign of the product and make calculations straightforward:

• The product of two positive numbers is always positive.
• When you multiply a positive number by a negative number, the result is always negative.
• The product of two negative numbers is positive. This might seem counter-intuitive at first, but think of it as "two negatives make a positive." This is because the negative signs essentially "cancel out."

By applying these rules, you can quickly determine the sign of your answer without having to rely on trial and error.
For example, in multiplying (-7) and (-7):

• Recognize both numbers are negative.
• Use the rule that the product of two negative numbers is positive, resulting in a positive answer.

Knowing these multiplication rules will help you confidently approach mathematical problems involving positive and negative numbers.

Absolute value

The absolute value of a number is the distance of that number from zero on the number line, regardless of direction. It's always a non-negative number. When dealing with multiplication, absolute value becomes helpful in simplifying calculations.

• For example, the absolute value of both (-7) and 7 is 7. This is because they are each 7 units away from zero.
• The absolute value is denoted by two vertical bars, like this: | -7 | = 7.
• Using absolute values, you can focus on multiplying only the magnitudes of the numbers involved, without worrying about their signs.

In our example, multiply the absolute values: 7 times 7 to get 49. This step ensures the product of the numbers without the influence of their sign.
Remember to apply the multiplication rules after calculating the absolute value to determine the final sign of the answer.