Question

Find the next three perfect squares after \(16 .\)

Short answer

The next three perfect squares are 25, 36, and 49.

Step-by-step solution

Understanding Perfect Squares

A perfect square is a number that can be expressed as the product of an integer with itself. For example, 16 is a perfect square because it is the product of 4 multiplied by 4, i.e., \(4^2 = 16\).

Identify the Next Integer

Since 16 is a perfect square (\(4^2\)), we need to find the perfect squares of the next three consecutive integers. The next integer after 4 is 5.

Calculate the Next Perfect Square

To find the perfect square of 5, calculate \(5^2\). This is done by multiplying 5 by itself: \(5 \times 5 = 25\). So, 25 is the next perfect square after 16.

Repeat for the Next Integer

The next integer after 5 is 6. Calculate the perfect square of 6: \(6^2 = 6 \times 6 = 36\). Hence, 36 is the second perfect square after 16.

Find the Third Perfect Square

The following integer is 7. Calculate the perfect square of 7: \(7^2 = 7 \times 7 = 49\). Thus, 49 is the third perfect square after 16.

Key concepts

Understanding Integers

Integers are whole numbers that can be either positive, negative, or zero. They are fundamental in mathematics as they are used in a variety of operations and calculations, especially when exploring number properties like perfect squares.
In the context of perfect squares, integers are the base numbers that are multiplied by themselves. For instance, in the example of the number 16, the integer used is 4, since 4 multiplied by itself results in the perfect square 16.
Perfect squares themselves are specific integers, as each is the result of an integer being squared. This connection highlights the unique role integers play in the formation of perfect squares.

The Basics of Multiplication

Multiplication is a crucial mathematical operation often visualized as repeated addition. When you multiply two numbers, you are essentially adding one number to itself as many times as the value of the other number. For example, multiplying 5 by 5 can be seen as adding 5 five times, resulting in 25.
In relation to perfect squares, multiplication helps in determining what number, when multiplied by itself, gives a perfect square. As shown in our exercise, we found the perfect square of the integer 5 by multiplying it by 5:

• 5 × 5 = 25
• 6 × 6 = 36
• 7 × 7 = 49

This process of multiplying an integer by itself is central to proving a number is a perfect square.

Applying Mathematics Education Concepts

Mathematics education provides the foundation for understanding and solving problems involving integers and multiplication, such as finding perfect squares. One important concept taught is the idea of patterns and sequences, which help in predicting and calculating subsequent numbers in a series.
In the exercise, we applied these educational principles to find the next three perfect squares after 16 by identifying a pattern with integers. Each step involved using multiplication, which is a core arithmetic concept. As students progress in mathematics education, they learn to apply such logical processes to a wide range of problems, helping them understand and internalize mathematical principles.
By consistently engaging in exercises like these, students reinforce their numerical fluency and their ability to recognize patterns within integers, aiding their journey in mathematics education.