To determine the equivalent function for \( \frac{x+1}{x^{3x}} \), we can simplify it by examining each option and comparing it to the given function.
Let's start with option a) \( \frac{1}{x^2} - \frac{1}{x^3} \). The denominator in our original function is \( x^{3x} \), which cannot be factored into separate terms like \( x^2 \) and \( x^3 \). So option a) is not equivalent.
Next, let's consider option b) \( \frac{1}{x^3} - \frac{1}{x} \). This is closer to our original function. We can simplify option b) to \( \frac{x - x^3}{x^4} \). However, the denominator \( x^4 \) is not equivalent to \( x^{3x} \). So option b) is also not equivalent.
Moving on to option c) \( \frac{1}{x^2 - 1} \). This is a different type of function, as it includes subtraction in the denominator. Therefore, option c) is not equivalent.
Finally, let's consider option d) \( \frac{1}{x^2 - x} \). Similar to option c), this function includes subtraction in the denominator. So option d) is not equivalent.
Out of all the given options, we have determined that none of them (\( a \), \( b \), \( c \), \( d \), \( e \)) are equivalent to our original function \( \frac{x+1}{x^{3x}} \). Therefore, none of the options are correct.
Regarding the second question, a rational number is defined as a number that can be expressed as a fraction where the numerator and denominator are both integers. Let's analyze each option:
2: This is an integer and can be expressed as \( \frac{2}{1} \), so it is a rational number.
Pie (Ï€): Pi is an irrational number, meaning it cannot be expressed as a fraction, so it is not a rational number.
7: Like 2, this is an integer and can be written as \( \frac{7}{1} \), so it is a rational number.
5/25: This fraction can be simplified to \( \frac{1}{5} \), where the numerator and denominator are integers, so it is a rational number.
64/49: Again, this is a fraction with integers in both the numerator and denominator, so it is a rational number.
From the given options, the rational numbers are 2, 7, 5/25 (which can be simplified to 1/5), and 64/49.
Therefore, the answer to the second question is not 5/25 but rather 1/5.