For all x in the domain of the function x+1 over x^3x this function is equivalent to

\[a)\frac{1}{x^2}-\frac{1}{x^3}\\
b)\frac{1}{x^3}-\frac{1}{x}\\
c)\frac{1}{x^2-1}\\
d)\frac{1}{x^2-x}\\
e) \frac{1}{x^3}\]
This is one of the questions to my math test can someone tell if im right?
Is it D im very confused on this one

Which of the following is a rational number
All numbers are squared
2
Pie
7
5/25
64/49
Is it 5/25

For #1, please retype.

On this forum we write fractions in the form a/b ,
or something like (4x+6)/(x^2 - 5)
is your original function (x+1)/(x^3)*x ?
that trailing x looks like a typo

for your #2
did you mean "All numbers are under a squareroot" ?
In that case √(64/49) = 8/7 , all others are irrational

Just massage things to see. For example,

\frac{1}{x^2}-\frac{1}{x^3} = 1/x^2 - 1/x^3
= x/x^3 - 1/x^3
= (x-1)/x^3

Play around with the others, and it should become clear.

D is clearly wrong. 1/(x^3-x) = 1/ [x(x^2-1)]
there's no way you're going to get that to a fraction with only x^3 in the bottom.

To determine the equivalent function of \(\frac{x+1}{x^3x}\), we can simplify the expression.

Step 1: Combine the terms in the numerator as \(x+1\).

Step 2: In the denominator, simplify \(x^3x\) to \(x^4\).

Therefore, the expression becomes \(\frac{x+1}{x^4}\).

None of the options provided match this simplified expression. So, none of the options (a), (b), (c), (d), or (e) are correct.

Regarding your second question, to identify which of the given numbers is rational, we need to confirm if it can be expressed as the quotient of two integers.

Among the options provided:
2 is rational because it can be written as \(\frac{2}{1}\).
π (pi) is irrational since it cannot be expressed as the ratio of two integers.
7 is rational because it can be written as \(\frac{7}{1}\).
\(\frac{5}{25}\) can be simplified to \(\frac{1}{5}\), which is rational.
\(\frac{64}{49}\) is rational because it can be simplified to \(\frac{8}{7}\), where both numerator and denominator are integers.

Therefore, the rational numbers among the given options are 2, 7, \(\frac{1}{5}\), and \(\frac{8}{7}\). None of the options match \(5/25\), so it is not a rational number.

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 (π): 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.