x is a number between 3 and 50.
How many vales of x is radical x / 13 a rational number?
"x is a number between 3 and 50." requires any number x (integer, fractional or decimal) between the given limits such that √(x/13) is rational.
The number of integers is limited to 1, namely 13, since √(13/13)=1 is rational. Multiplying 13 by the nearest perfect square exceeds the upper limit.
However, there are many non-integer rational numbers that fit the bill.
Basically, we multiply 13 by fractions whose numerator and denominator are perfect squares.
13/4
13*4/9
13*9/4
13*25/9
13*9/25
13*25/36
13*36/25
13*36/49
13*49/36
13*49/64
13*64/49
...
As you can see, the list can go on and on, and the list is long, if not infinite.
You may want to double check if the question actually meant:
"x is an integer between 3 and 50."
for √x/13 to be rational, x must be a perfect square.
perfect squares between 3 and 50 are
4,9,16,25,36, and 49
To solve this problem, we need to identify the values of x for which the expression (radical x / 13) is a rational number.
First, let's understand what a rational number is. A rational number is a number that can be expressed as a fraction, where the numerator and denominator are both integers. In other words, a rational number can be written in the form p/q, where p and q are integers and q is not equal to 0.
Now, let's analyze the expression (radical x / 13). The square root of x (√x) can also be represented as x^(1/2). So, the expression becomes (x^(1/2) / 13).
To determine if this expression is rational, we need to check if x^(1/2) is a perfect square. If it is, then the expression will be rational.
From the given information, we know that x is a number between 3 and 50. We need to find the values of x for which x^(1/2) is a perfect square.
To do this, we can start by listing the perfect squares between 3 and 50: 4, 9, 16, 25, 36, and 49.
So, there are six values of x (4, 9, 16, 25, 36, and 49) for which x^(1/2) is a perfect square. Therefore, there are six values of x for which the expression (radical x / 13) is a rational number.