Problem 87 - Prealgebra

Question

Expand and multiply. $$5^{2}$$

Step-by-Step Solution

Answer

The expanded result of \(5^2\) is 25.

Step 1: Understand Exponents

Before expanding, let's understand what the exponent means. When we see \(5^2\), it indicates 5 multiplied by itself. So \(5^2 = 5 \times 5\).

Step 2: Perform the Multiplication

Now that we understand \(5^2\), we perform the multiplication: \(5 \times 5 = 25\).

Step 3: Final Result

The expanded and multiplied form of \(5^2\) is 25. Therefore, \(5^2 = 25\).

Key Concepts

Multiplication

At its core, multiplication is a mathematical operation where one number, known as the multiplier, is added to itself a specific number of times. In the context of the exercise given, multiplication is essential in solving problems involving exponents.
Exponents like in the term \(5^2\) tell us how many times the base (which is 5 in this example) should be used in a multiplication.
This means \(5^2\) is interpreted as \(5 \times 5\), essentially multiplying the base number 5 by itself once more. Hence, the multiplication process is straightforward: add the base number to itself repeatedly based on the exponent value.

• The equation \(5 \times 1 = 5\) is simple multiplication.
• Expanding this to \(5 \times 5\) gives us the result 25.

Understanding multiplication and how it relates to exponents is crucial, as it gives meaning to terms like "squared" or "cubed" when working with numbers.

Expanded Form

Expanded form is a way of breaking down numbers or expressions to show the value of each individual part more clearly. When dealing with exponential expressions, using expanded form can be helpful for visual learners as it allows us to see the multiplication process.
For instance, in \(5^2\), writing it in expanded form translates to \(5 \times 5\).
This step of expanding gives a clearer picture of what the exponent indicates, especially with numbers larger than 2. It simplifies the understanding of how exponents work by effectively laying out the multiplication process as:

• Start with the base number (5 in our case).
• Repeat the multiplication based on the power indicated by the exponent.

This form is particularly useful in educational settings where students are beginning to learn about exponents, as it emphasizes the repeated multiplication feature of the base number according to the exponent.

Squared Numbers

Squared numbers are a special category of exponents where the exponent is specifically 2. This means that a number is multiplied by itself once.
For example, when we say \(5^2\), it refers to 5 being multiplied by itself, which equates to \(5 \times 5\).
When numbers are squared, they are often considered part of the "area" model in mathematics. Squaring a number gives us the area of a square with sides equal to the base number. It's a useful concept, especially in geometry:

• The term "squared" is derived from the shape of a square, where multiplying the side by itself gives its area.
• \(5^2\) illustrates that the area of a square with side length 5 is 25.

Recognizing squared numbers in mathematical expressions is key to grasping more complex algebraic and geometric concepts later on. Understanding simple exercises like \(5^2\) lays a strong foundation for more advanced studies in these areas.