Question
Divide and then simplify the rational expression. State any restrictions on the variable.
2x^4/3y^3 divided by 8y^-4/3x^-6
2x^4/3y^3 divided by 8y^-4/3x^-6
Answers
Answered by
oobleck
where did those fractional powers come from?
2x^4/3y^3 divided by 8y^-4/3x^-6
= 2x^4/3y^3 * 3x^-6/8y^-4
= 2x^4/3y^3 * 3y^4/8x^6
= 6x^4y^4/24x^6y^3
= y/4x^2
x and y cannot be zero
2x^4/3y^3 divided by 8y^-4/3x^-6
= 2x^4/3y^3 * 3x^-6/8y^-4
= 2x^4/3y^3 * 3y^4/8x^6
= 6x^4y^4/24x^6y^3
= y/4x^2
x and y cannot be zero
Answered by
GPT 3.5
To divide the rational expression, we need to simplify the rational fractions and then divide them.
First, let's simplify the expressions individually.
For the numerator: 2x^4 / 3y^3
The numerator is already simplified, so no further simplification is needed.
For the denominator: 8y^(-4/3) / x^(-6)
We can simplify the denominator by moving negative exponents to the numerator and changing their signs.
8y^(-4/3) / x^(-6) = 8 / (y^(4/3) * x^6)
Now that we have simplified the expressions, we can divide them by multiplying the first fraction by the reciprocal of the second fraction.
(2x^4 / 3y^3) / (8 / (y^(4/3) * x^6))
To divide, we multiply the first fraction by the reciprocal of the second fraction.
(2x^4 / 3y^3) * ((y^(4/3) * x^6) / 8)
When multiplying rational expressions, we can simplify by canceling common factors between the numerators and denominators.
Canceling factors:
(2x^4 / (3y^3)) * ((y^(4/3) * x^6) / 8)
= (2x^4 * y^(4/3) * x^6) / (3y^3 * 8)
To simplify further, we can combine like terms in the numerator and denominator.
= (2 * x^4 * x^6 * y^(4/3)) / (3 * 8 * y^3)
= (2 * x^(4 + 6) * y^(4/3)) / (24 * y^3)
= (2 * x^10 * y^(4/3)) / (24 * y^3)
Simplifying the expression gives us:
(x^10 * y^(4/3)) / (12 * y^3)
The restrictions on the variables are:
1. The variable y cannot equal zero since it appears in the denominator.
2. The variable x does not have any restrictions.
First, let's simplify the expressions individually.
For the numerator: 2x^4 / 3y^3
The numerator is already simplified, so no further simplification is needed.
For the denominator: 8y^(-4/3) / x^(-6)
We can simplify the denominator by moving negative exponents to the numerator and changing their signs.
8y^(-4/3) / x^(-6) = 8 / (y^(4/3) * x^6)
Now that we have simplified the expressions, we can divide them by multiplying the first fraction by the reciprocal of the second fraction.
(2x^4 / 3y^3) / (8 / (y^(4/3) * x^6))
To divide, we multiply the first fraction by the reciprocal of the second fraction.
(2x^4 / 3y^3) * ((y^(4/3) * x^6) / 8)
When multiplying rational expressions, we can simplify by canceling common factors between the numerators and denominators.
Canceling factors:
(2x^4 / (3y^3)) * ((y^(4/3) * x^6) / 8)
= (2x^4 * y^(4/3) * x^6) / (3y^3 * 8)
To simplify further, we can combine like terms in the numerator and denominator.
= (2 * x^4 * x^6 * y^(4/3)) / (3 * 8 * y^3)
= (2 * x^(4 + 6) * y^(4/3)) / (24 * y^3)
= (2 * x^10 * y^(4/3)) / (24 * y^3)
Simplifying the expression gives us:
(x^10 * y^(4/3)) / (12 * y^3)
The restrictions on the variables are:
1. The variable y cannot equal zero since it appears in the denominator.
2. The variable x does not have any restrictions.
Answered by
GPT 3.5
You are correct, I apologize for the error in my previous response. The correct simplification of the rational expression is:
2x^4/3y^3 divided by 8y^(-4/3)x^(-6)
= 2x^4/3y^3 * 3y^(4/3)x^6/8
= 6x^10y^4/24x^6y^3
= x^4y/4y^2x^2
= y/4x^2
The restrictions on the variables are:
1. The variable y cannot equal zero since it appears in the denominator.
2. The variable x cannot be zero since it appears in the denominator.
2x^4/3y^3 divided by 8y^(-4/3)x^(-6)
= 2x^4/3y^3 * 3y^(4/3)x^6/8
= 6x^10y^4/24x^6y^3
= x^4y/4y^2x^2
= y/4x^2
The restrictions on the variables are:
1. The variable y cannot equal zero since it appears in the denominator.
2. The variable x cannot be zero since it appears in the denominator.