Problem 21 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. $$ 4^{3}+9 \cdot 2 $$

Step-by-Step Solution

Answer

The evaluated expression equals 82

Step 1: Evaluate the exponent

The first operation to perform according to the order of operations, PEMDAS, is exponents. So \(4^3\) will be evaluated first. \(4^3 = 4 \cdot 4 \cdot 4 = 64\). So the expression becomes \(64 + 9 \cdot 2\)

Step 2: Carry out the multiplication

The next operation according to PEMDAS is multiplication. Perform the operation \(9 \cdot 2\), which equals 18. Substitute this value back into the expression. So the expression becomes \(64 + 18\)

Step 3: Perform the addition

The last operation according to PEMDAS is addition. Evaluate \(64 + 18\), which equals 82. Substitute this value back into the expression. Therefore, the evaluated expression equals 82

Key Concepts

Understanding Exponents

Exponents are a crucial part of mathematics, essential for representing repeated multiplication of a number by itself. When you see something like \(4^3\), it means that you multiply 4 by itself a total of three times. So, \(4^3\) is calculated as:

• Step 1: Begin with the base number, which is 4.
• Step 2: Multiply 4 by itself once (\(4 \times 4 = 16\)).
• Step 3: Multiply the result by 4 again (\(16 \times 4 = 64\)).

Thus, \(4^3 = 64\). Exponents simplify large or complicated multiplications, making calculations more efficient. They follow specific rules that fit into the larger order of operations in mathematics. Always remember, in expressions with mixed operations, exponents take precedence over multiplication and addition, making them your first step.

The Role of Multiplication

Multiplication is a fundamental arithmetic operation representing repeated addition. It appears in expressions as a process of adding a number to itself a specified number of times. Following the calculation of exponents, multiplication is often the next step, per the order of operations.For example, in the expression \(9 \cdot 2\):

• Start by identifying the numbers: 9 and 2.
• Multiply 9 by 2, meaning you add 9 two times (\(9 + 9 = 18\)).

The result of \(9 \cdot 2\) is 18. When solving mathematical problems, ensure you handle multiplication before proceeding to addition if both appear in the same expression. This ensures your calculations follow the correct order and produce accurate results.

Addition in the Order of Operations

Addition is one of the most basic arithmetic operations, where you combine numbers to get a sum. It's the last operation to conduct within the sequence known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). In our example, after handling exponents and multiplication, addition comes into play with the expression \(64 + 18\):

• First, evaluate the expression by adding the two numbers sequentially.
• Add 18 to 64 (64 + 18 = 82).

This operation combines the values you've previously calculated, leading to the final result. Addition wraps up your calculations, providing a complete solution to the problem at hand once all previous components are accurately processed.