Divide and then simplify the rational expression. State any restrictions on the variable.
2x^4/3y^3 divided by 8y^-4/3x^-6
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
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.
To divide and simplify the rational expression (2x^4/3y^3) divided by (8y^-4/3x^-6), we can follow these steps:
Step 1: Simplify the expression by flipping the second fraction and changing the division to multiplication:
(2x^4/3y^3) * (3x^-6/8y^-4)
Step 2: Simplify the numerator by multiplying the coefficients and adding the exponents:
(2 * 3 * x^4 * x^-6) = 6x^-2
Step 3: Simplify the denominator by multiplying the coefficients and adding the exponents:
(3 * 8 * y^3 * y^-4) = 24y^-1
So, the expression simplifies to:
6x^-2 / 24y^-1
Step 4: Simplify further by reducing the fraction:
Divide both the numerator and denominator by the greatest common factor, which is 6:
(6 / 6) * (x^-2 / 4y^-1) = 1 * (x^-2 / 4y^-1)
Simplifying further, we can rewrite x^-2 as 1/x^2, and y^-1 as 1/y:
1 * (1/x^2 * 1 / 4y)
Therefore, the simplified expression is:
(1 / 4xy).
Restrictions on the variable:
Since we have exponents in the denominator, we need to ensure that the denominators are not equal to zero. Hence, the restrictions are:
x ≠ 0, y ≠ 0.
To divide rational expressions, we follow these steps:
Step 1: Flip the second fraction (the divisor) and change the division sign to multiplication:
2x^4/3y^3 ÷ 8y^-4/3x^-6 becomes 2x^4/3y^3 * 3x^-6/8y^-4
Step 2: Simplify the expression:
For the numerator, multiply 2x^4/3y^3 by 3x^-6. This gives us (2x^4 * 3x^-6)/(3y^3).
For the denominator, multiply 8y^-4 by 3y^3. This gives us 8y^-4 * 3y^3.
Step 3: Simplify each part of the expression:
In the numerator, when we multiply x^4 by x^-6, we add the exponents since they have the same base: x^(4 + -6) = x^-2.
In the denominator, when we multiply y^-4 by y^3, we add the exponents since they have the same base: y^(-4 + 3) = y^-1.
Step 4: Write the simplified expression:
(2x^-2)/(3y^-1) * (3y^3)/(8y^-4).
Step 5: Combine the fractions and simplify:
(2x^-2 * 3y^3) / (3y^-1 * 8y^-4).
Multiplying the coefficients together, we get (2 * 3) = 6.
For x, combining x^-2 with x^-6, we add the exponents to get x^(-2 - 6) = x^-8.
For y, combining y^3 with y^-1 and y^-4, we add the exponents to get y^(3 - 1 - 4) = y^-2.
Therefore, the simplified expression is:
6x^-8/y^-2.
To clarify the restrictions on the variables:
There are no specific restrictions provided in the original expression. However, remember that division by zero is undefined, so any value of x or y that makes the denominator zero should be excluded from the domain.
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.