Problem 54 - Elementary Algebra

Question

Perform the operations. \(5 \cdot 2(-3)(-5)\)

Step-by-Step Solution

Answer

The result is 150.

Step 1: Multiply the Constants Inside Parentheses

First, compute the multiplication inside the parentheses: \((-3) \times (-5)\). To multiply two negative numbers, multiply their absolute values and take the result as positive, because the product of two negative numbers is positive. This results in 15.

Step 2: Multiply the Result by the Remaining Constants

Now, multiply the resulting number from Step 1 by 5 and 2: \(5 \times 2 \times 15\). First, multiply 5 and 2 to get 10, then multiply 10 by 15.

Step 3: Finalize the Multiplication

Calculate \(10 \times 15\). This equals 150, so the final result of the operations is 150.

Key Concepts

Negative Numbers

When dealing with negative numbers, it's important to remember some basic rules about multiplication and addition. In mathematics, a negative number is any number less than zero, depicted with a minus sign (-). They are typically used to represent loss, debt, or decrease.

When multiplying or dividing:

• Two negative numbers make a positive. For example, \((-3) imes (-5) = 15\) because both numbers are negative.
• A positive number by a negative number (or vice versa) results in a negative number. For example, \(5 imes (-3) = -15\).

Negative numbers change how operations are performed, and understanding the sign rules helps in various aspects of algebra and arithmetic.

Order of Operations

The order of operations is a fundamental rule in mathematics. It dictates the order in which different operations should be carried out to solve an equation correctly. This concept is crucial to avoid ambiguity in mathematical expressions, especially when multiple operations are involved.

The standard sequence is sometimes abbreviated as PEMDAS:

• Parentheses: Solve anything in parentheses first.
• Exponents: Compute powers or roots next.
• Multiplication and Division: From left to right, do these operations after parentheses and exponents.
• Addition and Subtraction: Finally, handle these last, also from left to right.

In the exercise given, you first multiply the numbers inside the parentheses (i.e., the negative numbers), then multiply that result by the other factors in the expression. This orderly method ensures accuracy in calculations.

Absolute Value

Absolute value is a way to express the magnitude of a number without regard to its sign. It is always non-negative and can be thought of as "how far a number is from zero on the number line."

The absolute value of a number \(x\) is written as \(|x|\). For instance:

• \(|-3| = 3\)
• \(|5| = 5\)

In multiplication, understanding that the product of two negative numbers is positive relies partly on acknowledging their absolute values. This insight helps clarify why, when two negatives are multiplied, the negative signs cancel out. Using absolute values simplifies handling such operations and prevents considering any confusing sign rules.