Problem 106 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. Without parentheses, an exponent has only the number next to it as its base.

Step-by-Step Solution

Answer

The statement 'Without parentheses, an exponent has only the number next to it as its base' makes sense and aligns with the mathematical order of operations.

Step 1: Understanding the concept

The exponent refers to the number of times a number, the base, is multiplied by itself. If there are no brackets involved, the number right next to the exponent symbol is considered as the base by default. This is consistent with the order of operations in mathematics, which generally follows the rule of BODMAS/BIDMAS/PEMDAS (Bracket, Order or power or exponent, Division, Multiplication, Addition and Subtraction). Within this rule, the operations are processed from left to right.

Step 2: Applying the concept to an example

Consider an example \(4^2 + 3\). Here, the exponent applies to the base '4' not to the '(4+3)'. So it is evaluated as \((4^2) + 3 = 16 + 3 = 19\) and not as \(7^2 = 49\). This example proves the statement.

Step 3: Conclusion

Therefore, the given statement 'Without parentheses, an exponent has only the number next to it as its base' does make sense because it is in line with the order of operations. Parentheses alter this order, causing the operations within them to be carried out first, but without parentheses, an exponent applies only to the number (base) immediately preceding it.

Key Concepts

Order of Operations

In mathematics, the order of operations is a fundamental principle that determines the sequence in which different operations are conducted. This is vital when evaluating expressions to ensure consistent and accurate results. The common rule to remember is PEMDAS or BIDMAS, which stands for:

• P: Parentheses
• E: Exponents (or Orders)
• M/D: Multiplication and Division (from left to right)
• A/S: Addition and Subtraction (from left to right)

The rule dictates that expressions within parentheses are evaluated first, followed by any exponents or orders. Next, you perform multiplication and division as they appear from left to right. Finally, you handle addition and subtraction as they occur from left to right.

For instance, in the expression \(4^2 + 3\), you first calculate the exponent \(4^2\) given the order of operations rule, yielding \(16\), before adding \(3\) to the result.

Mathematical Expressions

Mathematical expressions are symbols and numbers combined to represent a particular value or relationship. Unlike equations, they do not have an equals sign but are used extensively to convey information about mathematical relationships. These expressions may include multiple operations such as addition, subtraction, multiplication, division, and exponents.

Understanding how to interpret mathematical expressions is critical because each symbol and number holds significance. Each operation must be performed in the correct order to obtain the accurate result. Mathematical expressions can vary greatly in complexity and can take many forms. The simple expression \(4^2 + 3\) combines an exponent operation and an addition operation. It's crucial for students to practice determining which numbers and operations are grouped by any parentheses to correctly evaluate expressions.

Mastering the interpretation of mathematical expressions is essential in solving problems accurately and efficiently.

Parentheses in Math

Parentheses play a crucial role in determining the order in which operations are performed in mathematical expressions. They have the power to change the sequence by which numbers and operations are processed, impacting the outcome of calculations.

When parentheses are included in an expression, everything inside them must be calculated first before dealing with other operations outside. This overrides the typical order of operations, making parentheses the highest priority. For example, in the expression \((4 + 3)^2\), the addition inside the parentheses is completed first, resulting in \(7\), before applying the exponent to get \(49\).

Without parentheses, operations are carried out strictly according to the conventional order of operations. This means in an expression like \(4^2 + 3\), the exponent is processed first, followed by the addition. Understanding how and where to use parentheses effectively can greatly benefit in evaluating expressions correctly.