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 is the square of some integer. Perfect squares are important in the context of decimals and simplifying radical expressions.
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The square of an integer is a perfect square. For example, 4 is a perfect square because it is the square of 2 (4 = 2^2).
Every perfect square has an integer square root. For instance, the square root of 9 is 3, because 3 x 3 = 9.
When a number is a perfect square, its square root can be expressed without a radical symbol. For example, the square root of 16 is simply 4.
Identifying perfect squares is crucial for simplifying radical expressions by removing the radical symbol when possible.
The decimal representation of a perfect square is a terminating decimal, meaning it has a finite number of digits after the decimal point. For example, the decimal representation of 4 is 2.0.
Review Questions
Explain how the concept of a perfect square is related to the decimal representation of a number.
The decimal representation of a perfect square is a terminating decimal, meaning it has a finite number of digits after the decimal point. This is because the square root of a perfect square is an integer, and when that integer is multiplied by itself, the result is the original perfect square. For example, the square root of 4 is 2, and 2 multiplied by 2 is 4, which has a decimal representation of 2.0.
Describe how the identification of perfect squares can be used to simplify radical expressions.
When a number under a radical symbol is a perfect square, the radical can be removed, and the square root can be expressed as an integer. This is because the square root of a perfect square is an integer. For example, the square root of 16 can be simplified to 4, because 16 is a perfect square (4^2 = 16). Identifying and factoring out perfect squares is a key step in simplifying more complex radical expressions.
Analyze the relationship between perfect squares, their square roots, and the properties of exponents.
The square root of a perfect square is an integer, and this integer can be expressed using the properties of exponents. Specifically, the square root of a perfect square $x^2$ is equal to $x$, because $x^2 = x \cdot x$. This relationship between perfect squares, their square roots, and the properties of exponents is fundamental to understanding and manipulating radical expressions, as well as working with decimal representations of numbers.
The square root of a number is the value that, when multiplied by itself, gives the original number. The square root of a perfect square is an integer.
A radical expression is an expression that contains a square root symbol. Simplifying radical expressions often involves identifying perfect squares.
Decimal Representation: The decimal representation of a perfect square is a terminating decimal, meaning it has a finite number of digits after the decimal point.