Question

If an expression without grouping symbols includes addition and an exponent, which operation should you do first?

Short answer

Exponents should be done first, before addition.

Step-by-step solution

Identifying the operations

In the exercise, two operations are mentioned: addition and exponent. No information is given about the presence of any other operations or grouping symbols.

Applying the order of operations

According to the order of operations (BIDMAS/PEMDAS), exponents are performed before addition. This order needs to be followed when there are no grouping symbols.

Key concepts

Exponents

The concept of exponents, also known as 'powers', is fundamental in mathematics. An exponent signifies how many times a number, the base, is multiplied by itself. For instance, in the expression \( 3^2 \) (read as 'three squared'), the number 3 is the base and the exponent is 2, indicating that 3 should be multiplied by itself once: \( 3^2 = 3 \times 3 = 9 \).

When tackling problems that involve both exponents and addition, it’s crucial to handle the exponents first. For example, in the expression \( 2 + 3^2 \), you would first calculate \( 3^2 \) to get 9, and then add that to 2, resulting in 11. Understanding the hierarchy of operations, which exponents are a part of, enables us to consistently reach the correct answer when simplifying mathematical expressions.

Addition

Addition is one of the basic arithmetic operations and serves as the foundation for much more complex math concepts. It involves combining two or more numbers to find their total. For example, if you add 2 and 3, the result is 5, expressed as \( 2 + 3 = 5 \).

While addition is straightforward, its position in the order of operations is important to understand. It occurs later in the sequence, after exponents have been resolved. This is to ensure that more complex parts of the expression are dealt with first, allowing for the components of the expression to be broken down logically.

BIDMAS

BIDMAS is an acronym used in some countries, like the UK, to remember the order of operations in mathematics: Brackets (or Parentheses), Indices (or Exponents), Division and Multiplication (from left to right), and Addition and Subtraction (from left to right).

This rule assists students in tackling mathematical expressions in a structured way. For example, given the expression \( 5 + 4^2 \times 3 \), we first address the Indices (Exponents), \( 4^2 = 16 \), followed by the Multiplication, \( 16 \times 3 = 48 \), and finally, we add 5 to obtain the result of 53. BIDMAS ensures you produce the correct answer by clarifying which operations take precedence over others.

PEMDAS

PEMDAS, also known as the order of operations, is an acronym used primarily in the United States to describe the sequence in which arithmetic operations should be performed to correctly solve expressions. It stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

Utilizing PEMDAS, let’s examine the expression \( (2 + 3) \times 2^3 \). We begin with the Parentheses, calculating \( 2 + 3 = 5 \). Next come the Exponents, \( 2^3 = 8 \), followed by Multiplication, \( 5 \times 8 = 40 \). PEMDAS is invaluable as a tool because it provides a consistent approach to solving mathematical problems, thus ensuring clarity and avoiding ambiguity in calculations.