Perfect Squares

Definition

A perfect square is a number that can be expressed as the product of two equal integers. In other words, a perfect square is the result of multiplying a number by itself. These numbers hold important properties and relationships within the context of real numbers and working with square roots.

5 Must Know Facts For Your Next Test

1. The first several perfect squares are: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.
2. Any number that is the square of an integer is a perfect square. For example, 9 is a perfect square because it is 3 × 3.
3. The square root of a perfect square is always a rational number, while the square root of a non-perfect square is an irrational number.
4. Perfect squares play an important role in simplifying and evaluating square root expressions.
5. When adding or subtracting square roots, it is important to identify any common perfect square factors to simplify the expression.

Review Questions

• Explain how perfect squares are related to the real number system and the properties of square roots.
  • Perfect squares are a subset of the real number system, specifically the rational numbers. This is because any number that is the square of an integer can be expressed as a ratio of two integers, making it a rational number. The square root of a perfect square is always a rational number, while the square root of a non-perfect square is an irrational number. This relationship between perfect squares, rational numbers, and square roots is crucial for understanding and working with real numbers, as well as simplifying and evaluating square root expressions.
• Describe how perfect squares are used to simplify square root expressions.
  • When simplifying square root expressions, it is important to identify any perfect square factors within the radicand (the number inside the square root symbol). By extracting the perfect square factors, the expression can be rewritten as the product of the square root of the perfect square and the square root of the remaining non-perfect square factor. This process allows for the simplification of square root expressions and the representation of irrational numbers in a more manageable form.
• Analyze the role of perfect squares in the addition and subtraction of square root expressions.
  • When adding or subtracting square root expressions, the presence of common perfect square factors can significantly simplify the process. By identifying any common perfect square factors between the terms, the square root expressions can be combined or subtracted more efficiently. This is because the square root of a perfect square is a rational number, which can then be factored out, leaving only the square root of the remaining non-perfect square factors. Understanding the properties of perfect squares and their relationship to square root expressions is crucial for successfully performing these algebraic operations.

Related terms

Square Root: The square root of a number is the value that, when multiplied by itself, equals the original number. The square root symbol is $\sqrt{}$.

Irrational Numbers: Irrational numbers are real numbers that cannot be expressed as a simple fraction. The square root of non-perfect square numbers are irrational.

Rational Numbers: Rational numbers are real numbers that can be expressed as a ratio of two integers. Perfect squares are a subset of rational numbers.