Is the Square Root of 64 a Rational Number?
Understanding Rational Numbers
Rational numbers are numbers that can be expressed as a ratio of two integers, with the denominator not being zero. In simpler terms, a rational number can be written in the form of a fraction, where both the numerator and denominator are integers. Examples include common fractions such as ( frac{1}{2} ), and more complex ones like ( frac{16}{2} ).The Square Root of 64
The square root of 64 is 8. Mathematically, this can be written as ( sqrt{64} 8 ). This is a perfect square, meaning that 8 times 8 equals 64, or ( 8^2 64 ).Proving 8 is a Rational Number
To prove that 8 is a rational number, we use the definition of a rational number. According to the definition, a number is rational if it can be expressed as the ratio of two integers where the denominator is not zero. In this case, 8 can be written as ( frac{8}{1} ). Both 8 and 1 are integers, and 1 is not zero. Therefore, 8 is a rational number.Example Equivalences:
- ( 8 frac{16}{2} ): Here, 16 is an integer and 2 is an integer, making this another valid representation of 8 as a rational number. - ( 8 frac{80}{10} ): Again, both 80 and 10 are integers, further confirming that 8 is a rational number.Both Positive and Negative Square Roots
When discussing the square root, it is important to consider both the positive (principal) root and the negative root. The negative root, -8, is also a rational number because it can be expressed as ( frac{-8}{1} ) or ( frac{8}{-1} ). Both -8 and 1 (or -1) are integers, which confirms that -8 is a rational number.Generalization:
Any square number has both a positive and a negative square root, and both of these roots are rational numbers. For example, the square roots of 64 are 8 and -8, and both are rational numbers. This is true for any perfect square. For instance, the square roots of 4 are 2 and -2, and both are rational.