Problem 110 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(109-116\), write a numerical expression for each phrase. Then simplify the numerical expression by performing the given operations. 14 added to the product of 3 and \(-15\)

Step-by-Step Solution

Answer

The simplified form of the given phrase is \(-31\).

Step 1: Interpret the Phrase into a Mathematical Expression

The given phrase, '14 added to the product of 3 and -15' can be translated into mathematical terms as \(14 + (3 \times -15)\)

Step 2: Perform the Multiplication

The first operation to be performed according to order of operations is multiplication, so multiply 3 by -15. This gives \(-45\). The expression now becomes \(14 + -45\) or \(14 - 45\).

Step 3: Perform the Addition

Perform the addition operation. The result of \(14 - 45\) is \(-31\).

Key Concepts

Understanding the Order of Operations

When solving numerical expressions, it's important to know the sequence in which to perform each mathematical operation. This sequence is known as the "Order of Operations" and is crucial to obtaining the correct answer.

• Typically, the common mnemonic PEMDAS helps us remember the order: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
• In phrases like the one in our exercise—“14 added to the product of 3 and -15”—we can see that multiplication must be handled before addition.
• This ensures that numerical expressions are simplified correctly and that everyone arrives at the same result.

By adhering to the order of operations, we can effectively solve expressions like the one given, ensuring that each step leads logically to the next until the calculation is complete.

Exploring Multiplication

Multiplication is one of the core operations in mathematics. It serves to combine equal groups of items or, in more abstract terms, it's a way to scale one number by another.

• In our scenario, multiplication is used to find the product of 3 and exted{-15}. The term "product" refers specifically to the result of a multiplication operation.
• The calculation of multiplication first transforms our expression from this \(14 + (3 \times -15)\) into a simpler form, by resolving the terms within the parenthesis.
• For the example expression, multiplying 3 by -15 results in -45, as a positive number multiplied by a negative number yields a negative result.

Mastering multiplication is essential in simplifying numerical expressions accurately and efficiently, especially when dealing with negative numbers.

Addition and Subtraction in Expressions

Addition and subtraction are typically among the final steps in the order of operations. They are used to combine or adjust quantities, and they must be performed from left to right when they occur together.

• In our example expression \(14 - 45\), addition is essentially taking 14 and removing 45 from it.
• This can sometimes be rephrased as "finding the difference" between the two numbers.
• The result of this operation is -31, indicating that after removing 45 from 14, we are left with a negative balance.

Understanding how to correctly apply addition and subtraction rules is vital, especially when these operations involve negative numbers. It ensures accuracy in calculating numerical expressions and allows us to express real-world scenarios mathematically.