Determine if √8 is rational or irrational and give a reason for your answer.
√8 is irrational.
To determine if a number is rational or irrational, we need to consider its decimal representation. If the decimal representation is non-repeating and non-terminating, then the number is irrational.
√8 can be simplified as √(4 * 2) = √4 * √2 = 2 * √2.
Since √2 is irrational (proved by Pythagoras' theorem), 2 * √2 is also irrational.
Therefore, √8 is irrational.
To determine if √8 is rational or irrational, we first need to recall the definitions of rational and irrational numbers.
A rational number is a number that can be expressed as the quotient (or fraction) of two integers where the denominator is not zero.
An irrational number is a number that cannot be expressed as a fraction of two integers.
Now let's determine if √8 is rational or irrational.
√8 can be simplified as follows:
√8 = √(4 × 2)
Since 4 is a perfect square, it can be factored out of the square root:
√8 = √4 × √2
Simplifying further:
√8 = 2√2
Now, we can see that √8 is expressed as the product of 2 (a rational number) and √2 (an irrational number).
Therefore, √8 is an irrational number.
To determine if √8 is rational or irrational, we need to understand the definitions of rational and irrational numbers.
A rational number is any number that can be expressed as a fraction of two integers, where the denominator is not zero. In other words, it can be written as a/b, where a and b are both integers and b is not equal to zero.
An irrational number, on the other hand, cannot be expressed as a fraction of two integers. It is a non-terminating, non-repeating decimal.
To find out if √8 is rational or irrational, let's simplify the square root of 8.
√8 = √(4 * 2) = √4 * √2 = 2√2
Now we can see that √8 can be expressed as 2√2. Since 2 and √2 are both rational numbers (2 is an integer and √2 can be expressed as a fraction √2/1), we can conclude that √8 can be expressed as the product of two rational numbers.
Therefore, √8 is a rational number.