Perfect Square

Definition

A perfect square is a number that can be expressed as the product of two equal integers. In other words, it is a number that can be obtained by multiplying a whole number by itself. Perfect squares are important in the context of simplifying and using square roots, as they have unique properties that can be leveraged in mathematical operations.

5 Must Know Facts For Your Next Test

1. The first few perfect squares are: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.
2. Any perfect square can be expressed as the product of two equal integers, such as 9 = 3 × 3.
3. The square root of a perfect square is always a whole number, such as √16 = 4.
4. Perfect squares have unique properties that can be used to simplify square root expressions, such as $\sqrt{100} = 10$.
5. Identifying perfect squares is important in the context of simplifying and using square roots, as they can be used to rewrite expressions in a more manageable form.

Review Questions

• Explain how perfect squares are related to square roots and radical notation.
  • Perfect squares are closely related to square roots and radical notation because they represent numbers that can be expressed as the product of two equal integers. This means that the square root of a perfect square will always be a whole number, which can be used to simplify expressions involving square roots. For example, $\sqrt{100}$ can be rewritten as 10, since 100 is a perfect square (10 × 10 = 100). This ability to simplify square root expressions using perfect squares is a key concept in the context of 5.7 Simplify and Use Square Roots.
• Describe how perfect squares can be used to simplify square root expressions.
  • Perfect squares can be used to simplify square root expressions by identifying factors within the expression that are perfect squares. For instance, if you have the expression $\sqrt{72}$, you can rewrite it as $\sqrt{(4 \times 18)}$, since 4 is a perfect square (2 × 2 = 4). This allows you to simplify the expression further to $2\sqrt{18}$, as the square root of 4 is 2. This process of identifying and factoring out perfect squares is a crucial skill in the context of 5.7 Simplify and Use Square Roots, as it enables you to work with square root expressions more efficiently.
• Analyze how the properties of perfect squares can be applied to solve problems involving square roots.
  • The properties of perfect squares can be applied to solve a wide range of problems involving square roots in the context of 5.7 Simplify and Use Square Roots. For example, if you need to evaluate the expression $\sqrt{144 + 81}$, you can first identify that 144 and 81 are both perfect squares (12^2 and 9^2, respectively). This allows you to rewrite the expression as $\sqrt{(12^2) + (9^2)}$, which can then be simplified to $\sqrt{225} = 15$. By recognizing and leveraging the properties of perfect squares, you can transform complex square root expressions into more manageable forms, making it easier to solve problems and demonstrate your understanding of the concepts covered in this chapter.

Related terms

Square Root:

The square root of a number is the value that, when multiplied by itself, produces the original number. For example, the square root of 16 is 4, because 4 × 4 = 16.

Radical Notation: Radical notation is a way of representing square roots using the radical symbol (√). For example, the square root of 9 can be written as √9.

Exponents:

Exponents are a way of representing repeated multiplication of a number. For example, 4^2 is the same as 4 × 4, which is a perfect square.