Determine if each root is a rational or irrational number. Explain your reasoning.
√36
3√78
√x is rational if x is a perfect square.
otherwise, not.
For example, √25 is rational because 25 = 5*5 = 5^2
Determine if the root is a rational or irrational number. Explain your reasoning.
√10
To determine if a root is a rational or irrational number, we need to understand the concept of rational and irrational numbers.
Rational numbers are numbers that can be expressed as a fraction of two integers, where the denominator is not zero. Irrational numbers, on the other hand, cannot be expressed as a fraction and have non-repeating, non-terminating decimal representations.
Let's apply this understanding to the given roots:
1. √36:
First, we simplify the square root of 36. The square root of 36 is 6 because 6 * 6 = 36. In this case, 6 is a whole number, which means it can be expressed as a fraction, such as 6/1. Since it can be expressed as a fraction, √36 is a rational number.
2. 3√78:
To determine if 3√78 is a rational or irrational number, we need to simplify √78. The square root of 78 cannot be simplified to a whole number, which indicates that it is an irrational number. Multiplying an irrational number by a rational number (in this case, 3) does not change its irrationality. Therefore, 3√78 is an irrational number.
In summary, √36 is a rational number, and 3√78 is an irrational number.