Write each rational expression in simplest form. State all restrictions on the variables.
48๐โ3๐^3/2๐^2+2๐โ8a
To simplify the rational expression 48๐โ3๐^3/2๐^2+2๐โ8a, we can factor out the greatest common factor in the numerator, which is 3๐:
3๐(16โ๐^2)/2๐^2+2๐โ8a
Next, we can factor out the greatest common factor in the denominator, which is 2๐:
3๐(16โ๐^2)/2๐(๐+1)โ4(๐+1)
Now, we can cancel out the common factors (3๐)/(2๐):
(16โ๐^2)/(๐+1)โ2(๐+1)
The simplified form of the rational expression is (16โ๐^2)/(๐+1)โ2(๐+1).
Now, let's state the restrictions on the variables. In the original expression, the denominator 2๐^2+2๐โ8a cannot be equal to zero since division by zero is undefined. Therefore, we can solve the quadratic equation 2๐^2+2๐โ8a=0 to find the restrictions.
Using the quadratic formula, we get:
๐ = (-2 ยฑ โ(2^2-4(2)(-8a)))/(2(2))
Simplifying the equation:
๐ = (-2 ยฑ โ(4+64a))/(4)
The restrictions on the variable ๐ are the values that make the denominator of the original expression zero. Thus, the restrictions are ๐ = (-2 + โ(4+64a))/(4) and ๐ = (-2 - โ(4+64a))/(4).