Add:
(6x)/(5z^(2))+(2y)/(xz)+(z)/(xy^(2))
put evefrything over a common denominator of 5x y^2 z^2 to get
(6x^2y^2 + 10y^3z + 5z^3) / 5xy^2z^2
To add the given terms:
(6x)/(5z^2) + (2y)/(xz) + (z)/(xy^2)
First, let's find the least common denominator (LCD) for the fractions. The LCD is the smallest multiple of all the denominators. In this case, the denominators are z^2, xz, and xy^2.
To determine the LCD, we need to factorize the denominators and find the highest power for each common factor.
The denominators can be factorized as follows:
z^2 = z * z
xz = x * z
xy^2 = x * y * y
From the factorization, we can see that the highest power of z is z^2, for x it's x, and for y it's y^2.
Therefore, the LCD is z^2 * x * y^2.
Now, we need to multiply each term by the appropriate factor to convert the denominator to the LCD.
For the first term: (6x)/(5z^2)
To convert the denominator z^2 to the LCD z^2 * x * y^2, we multiply the numerator and denominator by x * y^2.
(6x * x * y^2) / (5z^2 * x * y^2) = (6x^2y^2) / (5xzy^2)
For the second term: (2y)/(xz)
To convert the denominator xz to the LCD, we multiply the numerator and denominator by y * y^2.
(2y * y * y^2) / (xz * y * y^2) = (2y^4) / (xy^3z)
For the third term: (z)/(xy^2)
To convert the denominator xy^2 to the LCD, we multiply the numerator and denominator by z^2 * x.
(z * z^2 * x) / (xy^2 * z^2 * x) = (z^3x) / (xy^2z^2)
Now, we can add the fractions together:
(6x^2y^2) / (5xzy^2) + (2y^4) / (xy^3z) + (z^3x) / (xy^2z^2)
To simplify, let's combine the denominators by multiplying them together:
(6x^2y^2 * xy^3z * xy^2z^2 + 2y^4 * z^2 * x * xy^2z^2 + z^3x * 5xzy^2 * xy^3z) / (5xzy^2 * xy^3z * xy^2z^2)
Simplifying further:
(6x^3y^5z^3 + 2y^4zx^2yz^2 + 5x^2y^2z^4) / (5x^2y^6z^4)
So, the sum of the given terms is (6x^3y^5z^3 + 2y^4zx^2yz^2 + 5x^2y^2z^4) / (5x^2y^6z^4).