$$ \frac{4}{z}+\frac{2}{y} $$ $$ \frac{3}{z}-\frac{9}{y} $$ $$ \frac{4}{z}+\frac{2}{y} $$ $$ \frac{3}{z}-\frac{9}{y} $$ Simplify. (Simplify your answer. Use integers or fractions for any numbers in the expression.)

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UpStudy AI · Accepted Answer

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)
The simplified expression is $$ \frac{36(2y + z)^2(y - 3z)^2}{y^4z^4} $$.

\( \frac{4}{z}+\frac{2}{y} \) \( \frac{3}{z}-\frac{9}{y} \) \( \frac{4}{z}+\frac{2}{y} \) \( \frac{3}{z}-\frac{9}{y} \) Simplify. (Simplify your answer. Use integers or fractions for any numbers in the expression.)

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