Question
\( \frac{4}{z}+\frac{2}{y} \) \( \frac{3}{z}-\frac{9}{y} \) \( \frac{4}{z}+\frac{2}{y} \) \( \frac{3}{z}-\frac{9}{y} \) Simplify. (Simplify your answer. Use integers or fractions for any numbers in the expression.)
Ask by Daniel Wang. in the United States
Feb 27,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The simplified expression is \( \frac{36(2y + z)^2(y - 3z)^2}{y^4z^4} \).
Solution
We are given the four factors
(4/z + 2/y)
(3/z – 9/y)
(4/z + 2/y)
(3/z – 9/y)
Since the first two factors are repeated, we can write the product as
[(4/z + 2/y)(3/z – 9/y)]².
Step 1. Combine the two factors.
Write each expression as a single fraction. For the first factor:
4/z + 2/y = (4y)/(yz) + (2z)/(yz) = (4y + 2z)/(yz).
For the second factor:
3/z – 9/y = (3y)/(yz) – (9z)/(yz) = (3y – 9z)/(yz).
Now multiply them:
(4y + 2z)/(yz) · (3y – 9z)/(yz) = [(4y + 2z)(3y – 9z)]/(y²z²).
Step 2. Factor common numbers in the numerator.
Notice that:
4y + 2z = 2(2y + z) and 3y – 9z = 3(y – 3z).
Thus the product becomes:
[2(2y + z) · 3(y – 3z)]/(y²z²) = [6(2y + z)(y – 3z)]/(y²z²).
Step 3. Square the result.
Since our original expression is the square of the above product, we have:
{[6(2y + z)(y – 3z)]/(y²z²)}²
= 36(2y + z)²(y – 3z)² / (y⁴z⁴).
This is the simplified form of the original expression.
Final Answer:
36(2y + z)²(y – 3z)² / (y⁴z⁴)
Answered by UpStudy AI and reviewed by a Professional Tutor
Like
error msg
Explain 
Simplify this solution Mind Expander
Let's first combine the two expressions: 1. The first expression is \( \frac{4}{z} + \frac{2}{y} \). 2. The second expression is \( \frac{3}{z} - \frac{9}{y} \). Now, we'll add them together: \[ \left(\frac{4}{z} + \frac{2}{y}\right) + \left(\frac{3}{z} - \frac{9}{y}\right) = \frac{4}{z} + \frac{3}{z} + \frac{2}{y} - \frac{9}{y} \] Combine like terms: \[ \frac{4 + 3}{z} + \frac{2 - 9}{y} = \frac{7}{z} - \frac{7}{y} \] So, the simplified expression is: \[ \frac{7}{z} - \frac{7}{y} \] Thus, the final answer is: \[ \frac{7}{z} - \frac{7}{y} \]
