If (y-2)hole power 3 = (y+2)hole power 3 then find the value of Y.
a) -16
b) -4
c) O
d) 4
If you substitute any of the four answer choices into the equation:
(y-2)^3=(y+2)^3
you will find that none of them is a good answer.
So I suspect there is a typo for the question.
The equation as shown above can be solved by assuming
y=a+bi
Expand both sides to give
12b^2-24abi-12a^2-16=0
from which b can be solved by assuming a=any number, including zero.
(y-2)^3 = (y+2)^3.
(y-2)^2(y-2) = (y+2)^2(y+2).
(y^2-4y+4)(y-2) = (y^2+4y+4)(y+2)
y^3-4y^2+4y-2y^2+8y-8=y^3+4y^2+4y+2y^2+8y+8,
-6y^2+12y-8 = 6y^2+12y+8,
-12y^2 = 16, Y = 1.155i.
No real solution, because y-2 cannot equal y+2.
To find the value of Y, we can solve the given equation step by step:
Step 1: Expand the given equation.
(y - 2)^3 = (y + 2)^3
(y - 2)(y - 2)(y - 2) = (y + 2)(y + 2)(y + 2)
Step 2: Simplify both sides of the equation.
(y - 2)(y - 2)(y - 2) = (y + 2)(y + 2)(y + 2)
(y - 2)^3 = (y + 2)^3
Step 3: Use the formula for the difference of cubes.
(a^3 - b^3) = (a - b)(a^2 + ab + b^2)
Applying the formula to the left side of the equation:
(y - 2)^3 = (y - 2)(y^2 + 2y + 4)
Applying the formula to the right side of the equation:
(y + 2)^3 = (y + 2)(y^2 + 2y + 4)
Step 4: Set the two expressions equal to each other.
(y - 2)(y^2 + 2y + 4) = (y + 2)(y^2 + 2y + 4)
Step 5: Expand and simplify both sides of the equation.
y^3 + 2y^2 + 4y - 2y^2 - 4y - 8 = y^3 + 2y^2 + 4y + 2y^2 + 4y + 8
Step 6: Combine like terms.
y^3 + 2y^2 - 2y^2 + 4y - 4y + 4y - 4y + 8 = y^3 + 2y^2 + 2y^2 + 4y + 4y + 4y + 8
Step 7: Simplify.
y^3 = y^3 + 8
Step 8: Subtract y^3 from both sides of the equation.
y^3 - y^3 = y^3 + 8 - y^3
0 = 8
Step 9: Since the equation 0 = 8 is not true, it means that there is no valid solution for Y that satisfies the given equation.
Therefore, the value of Y cannot be determined from the given equation. The correct answer is (c) 0.