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 =
1. Let's simplify the expression step-by-step:
First, let's factorize the denominators:
Y^2 - 9 = (Y + 3)(Y - 3)
3Y + 6 = 3(Y + 2)
2Y - 6 = 2(Y - 3)
Now, the expression becomes:
[(Y + 2)/(Y + 3)(Y - 3)] * [(4Y + 12)/(3(Y + 2))] ÷ [8/(2(Y - 3))]
Next, we can cancel out common factors between the numerators and denominators:
[(Y + 2)/(Y + 3)(Y - 3)] * [(4(Y + 3))/(3)] ÷ [8/(2(Y - 3))]
Simplifying further, we get:
[(Y + 2)/(Y + 3)(Y - 3)] * [4(Y + 3)/3] ÷ [4/(Y - 3)]
Now, cancel out the common factors again:
[(Y + 2)/(Y + 3)(Y - 3)] * [(Y + 3)] ÷ [1]
Finally, we simplify the expression to:
(Y + 2)/(Y - 3)
2. Let's simplify the expression step-by-step:
First, let's factorize the denominators:
X + Y = 1(X + Y)
16X^2 - Y^2 = (4X + Y)(4X - Y)
16X + 4Y = 4(4X + Y)
Now, the expression becomes:
[(4X - Y)/(X + Y)] * [(X^2 - Y^2)/(4X + Y)] ÷ [(2X - 2Y)/(4(4X + Y))]
Next, we can cancel out common factors between the numerators and denominators:
[(4X - Y)/(X + Y)] * [(X - Y)(X + Y)/1] ÷ [(2(X - Y))/(4(4X + Y))]
Simplifying further, we get:
[(4X - Y)/(X + Y)] * [(X - Y)/(1)] ÷ [2(X - Y)/(4(4X + Y))]
Now, cancel out the common factors again:
[(4X - Y)/(X + Y)] * [1] ÷ [2/(4(4X + Y))]
Finally, we simplify the expression to:
2X + 2Y
3. Let's simplify the expression step-by-step:
First, let's factorize the denominators:
2X + 12 = 2(X + 6)
4X - 20 = 4(X - 5)
X^2 + 7X + 10 = (X + 5)(X + 2)
8X = 8(X)
Now, the expression becomes:
[(X^2 - 25)/(2(X + 6))] * [(X^2 + 8X + 12)/(4(X - 5))] ÷ [(X + 5)(X + 2)/(8(X))]
Next, we can cancel out common factors between the numerators and denominators:
[(X - 5)(X + 5)/(2(X + 6))] * [(X + 6)(X + 2)/(4(X - 5))] ÷ [(X + 5)(X + 2)/(2(X))]
Simplifying further, we get:
[(X - 5)(X + 5)(X + 6)(X + 2)/(2(X + 6))] * [(X + 2)/(4(X - 5))] ÷ [(X + 5)(X + 2)/(2(X))]
Now, cancel out the common factors again:
[(X - 5)(X + 5)(X + 6)] * [1] ÷ [1/(2(X))]
Finally, we simplify the expression to:
2(X - 5)(X + 5)(X + 6)
To solve these arithmetic expressions, we can follow the order of operations, which is often remembered using the acronym PEMDAS:
1. Parentheses: Perform any calculations within parentheses first.
2. Exponents: Evaluate any exponents.
3. Multiplication and Division: Perform multiplication and division from left to right.
4. Addition and Subtraction: Perform addition and subtraction from left to right.
Let's apply these steps to the given expressions:
1. (Y+2)/(Y^2-9) * (4Y+12)/(3Y+6) ÷ 8/(2Y-6) =
First, apply the division operation within the first fraction:
(Y+2)/(Y^2-9) * (4Y+12)/(3Y+6) ÷ 8/(2Y-6) =
(Y+2)/(Y^2-9) * (4Y+12)/(3Y+6) * (2Y-6)/8
Next, simplify each fraction separately:
(Y+2)/(Y^2-9) = (Y+2)/((Y-3)(Y+3))
(4Y+12)/(3Y+6) = (4(Y+3))/(3(Y+2)) = 4/3
(2Y-6)/8 = (Y-3)/4
Now, substitute these simplified fractions back into the expression:
(Y+2)/((Y-3)(Y+3)) * 4/3 * (Y-3)/4
Next, cancel out common terms:
(Y+2) * (Y-3)/(Y-3) * 4/3 * (Y-3)/4 =
(Y+2) * 4/3
Finally, distribute and simplify:
4(Y+2)/3 = (4Y+8)/3
2. (4X-Y)/(X+Y) * (X^2-Y^2)/(16X^2-Y^2) ÷ (2X-2Y)/(16X+4Y) =
Let's simplify each fraction separately:
(4X-Y)/(X+Y) = (4X-Y)/(X+Y)
(X^2-Y^2)/(16X^2-Y^2) = (X-Y)(X+Y)/[(4X-Y)(4X+Y)]
(2X-2Y)/(16X+4Y) = 2(X-Y)/4(X+Y) = (X-Y)/(2(X+Y))
Now, substitute these fractions back into the expression:
(4X-Y)/(X+Y) * (X-Y)(X+Y)/[(4X-Y)(4X+Y)] ÷ (X-Y)/(2(X+Y))
Next, cancel out common factors:
(4X-Y)/(X+Y) * (X-Y)(X+Y)/[(4X-Y)(4X+Y)] * 2(X+Y)/(X-Y)
Now, simplify:
2(4X-Y)(X-Y)(X+Y)/(X+Y)(4X-Y)(4X+Y) * (X+Y)/(X-Y) =
2(X-Y)(X+Y)/(4X-Y)(4X+Y)
Finally, distribute and simplify:
2(X^2-Y^2)/(16X^2-Y^2) = 2(X^2-Y^2)/(16X^2-Y^2)
3. (X^2-25)/(2X+12) * (X^2+8X+12)/(4X-20) ÷ (X^2+7X+10)/8X =
Let's simplify each fraction separately:
(X^2-25)/(2X+12) = (X-5)(X+5)/2(X+6)
(X^2+8X+12)/(4X-20) = (X+2)(X+6)/4(X-5) = (X+2)(X+6)/4(X-5)
(X^2+7X+10)/8X = (X+5)(X+2)/8X
Now, substitute these fractions back into the expression:
(X-5)(X+5)/2(X+6) * (X+2)(X+6)/4(X-5) ÷ (X+5)(X+2)/8X
Next, cancel out common factors:
(X-5)(X+5)/2(X+6) * (X+2)(X+6)/4(X-5) * 8X/(X+5)(X+2)
Now, simplify:
8X(X-5)(X+5)(X+2)(X+6)/2(X+6)(4(X-5)(X+2)) * 8X/(X+5)(X+2)
Finally, cancel out common factors and simplify:
4X(X+6) = 4X^2 + 24X