Is the sqrt2 over 8 rational or irrational? I believe it is irrational.
I also need to know why.
your belief is correct
the reason is because √2 is irrational
... all non-integer roots of integers are irrational
I meant that it is the sqrt2 of 2 of 8 it is only the 2 that is sqrt2. This being said is what you say still true?
if this is what you mean ... (√2) / 8
then you're good
You are correct, the square root of 2 over 8 is irrational. To verify this, we should first understand what rational and irrational numbers are.
A rational number is any number that can be expressed as a fraction, where both the numerator and the denominator are integers, and the denominator is not zero. For example, 3/4, -2/5, and 7/1 are all rational numbers.
On the other hand, an irrational number is a number that cannot be expressed as a fraction of two integers. These numbers cannot be written as terminating or repeating decimals. Examples of irrational numbers include √2, π (pi), and e.
To determine if the square root of 2 over 8 is rational or irrational, let's simplify it first:
√2 / 8
We can rewrite the square root of 2 as 2^(1/2):
2^(1/2) / 8
Now, let's simplify further by breaking down 8 into its prime factorization:
2^(1/2) / (2^3)
Now, we can simplify the expression by subtracting the exponents:
2^(1/2-3) = 2^(-5/2)
Since the exponent is negative, we'll rewrite it as the reciprocal:
1 / (2^(5/2))
At this point, we still have an exponent in the denominator, indicating a square root in the expression. Since the expression contains a square root of 2, and it is not possible to express it as a fraction, the square root of 2 over 8 is irrational.
Therefore, your belief is correct. The square root of 2 over 8 is an irrational number.