1.(Y+2)/(Y^2-9) * (4Y+12)/(3Y+6) ÷ 8/(2Y-6) =
2.(4X-Y)/(X+Y) * (X^2-Y^2)/(16X^2-Y^2) ÷ (2X-2Y)/(16X+4Y) =
3.(X^2-25)/(2X+12) * (X^2+8X+12)/(4X-20) ÷ (X^2+7X+10)/8X =
To simplify these expressions, we can follow the order of operations (PEMDAS/BODMAS).
1. (Y+2)/(Y^2-9) * (4Y+12)/(3Y+6) ÷ 8/(2Y-6) =
First, let's simplify each sub-expression individually:
a. (Y+2)/(Y^2-9) can be simplified by factoring the denominator:
Y^2 - 9 = (Y+3)(Y-3)
So, (Y+2)/(Y^2-9) becomes (Y+2)/[(Y+3)(Y-3)].
b. (4Y+12)/(3Y+6) can be simplified by factoring out a 4 from both the numerator and the denominator:
(4Y+12)/(3Y+6) = 4(Y+3)/3(Y+2).
c. 8/(2Y-6) simplifies to 4/(Y-3) after dividing both the numerator and denominator by 2.
Now, substituting these simplified expressions back into the original expression:
[(Y+2)/[(Y+3)(Y-3)]] * [4(Y+3)/[3(Y+2)]] ÷ [4/(Y-3)].
To divide by a fraction, we can multiply by its reciprocal. Therefore, the expression can be rewritten as:
[(Y+2)/[(Y+3)(Y-3)]] * [4(Y+3)/[3(Y+2)]] * [(Y-3)/4].
Next, we can cancel out common factors from the numerators and denominators:
(Y+2) * (Y+3) * (Y+3) * (Y-3)
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[(Y+3) * (Y-3)] * [3(Y+2)] * [4].
As you can see, the (Y+3) terms and the (Y-3) terms cancel each other out. The expression simplifies to:
(Y+2) * (Y+3) * 3 / 4.
Now, we can multiply out the terms:
(Y^2 + 5Y + 6) * 3 / 4.
Finally, we can distribute the multiplication:
3Y^2 + 15Y + 18 / 4.
Therefore, the simplified expression is:
3Y^2 + 15Y + 18 / 4.
2. (4X-Y)/(X+Y) * (X^2-Y^2)/(16X^2-Y^2) ÷ (2X-2Y)/(16X+4Y) =
Following the same steps as in the previous example, we simplify each sub-expression individually and combine them at the end to get the simplified expression.
3. (X^2-25)/(2X+12) * (X^2+8X+12)/(4X-20) ÷ (X^2+7X+10)/8X =
Again, we will simplify each sub-expression individually and combine them at the end to get the simplified expression.