Problem 28 - Beginning and Intermediate Algebra

Question

Simplify each expression. \(2 \cdot 5^{2}\)

Step-by-Step Solution

Answer

The simplified expression is 50.

Step 1: Understand the Problem

We are given the expression \(2 \cdot 5^{2}\). The goal is to simplify this expression by performing the necessary arithmetic operations.

Step 2: Calculate the Exponent

First, calculate \(5^{2}\), which means that you multiply 5 by itself: \(5 \times 5 = 25\).

Step 3: Multiply the Result

Next, take the result from the exponentiation, which is 25, and multiply it by 2: \(2 \times 25 = 50\).

Key Concepts

Understanding Exponentiation

Exponentiation is a mathematical operation that involves raising a number, known as the base, to the power of another number, called the exponent. This operation tells us how many times to multiply the base by itself. In the expression \(5^2\), the base is 5 and the exponent is 2, which means we multiply 5 by itself once: \(5 \times 5\). This gives us 25.

• The exponent indicates repeated multiplication.
• It's a powerful tool to simplify complex multiplications.

When you see an exponent, the first step is to resolve it because it will transform into a simpler number, making further calculations easier.

Demystifying Arithmetic Operations

Arithmetic operations are the foundational building blocks of mathematics, comprising addition, subtraction, multiplication, and division. In the expression \(2 \cdot 5^2\), we focus on multiplication. Before diving into calculations, it's crucial to identify the type of operations required and in what order they need to be performed.

• Start by simplifying expressions with exponents.
• Follow by performing multiplication or other indicated operations.

By understanding the required operations, you can streamline the simplification process and avoid mistakes.

The Role of Multiplication

Multiplication is one of the primary operations used in arithmetic, represented by the symbol \(\cdot\) or \(\times\). It is essentially repeated addition. In our example, once we have simplified the exponent \(5^2\) to 25, we multiply the result by 2. Thus, the expression \(2 \cdot 25\) very simply translates to adding 25 two times: \(25 + 25 = 50\).

• Multiplication is used to quickly add equal groups of numbers.
• It follows the result of exponentiation in this context.

Completing multiplication after handling exponents allows us to efficiently arrive at the final simplified expression.

Importance of Order of Operations

Order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), is crucial in simplifying mathematical expressions. It dictates the sequence in which operations must be performed to achieve a correct result. In our case, \(2 \cdot 5^2\), it ensures that we first calculate the exponent before moving on to multiplication.

• ALWAYS perform calculations inside parentheses first, if any.
• Resolve exponents before multiplication and division.
• Conclude with addition and subtraction.

Following the order of operations correctly guarantees that expressions are simplified accurately every time.