To solve the equation \(\sqrt[4]{2 x-8}+\sqrt[4]{2 x+8}=0\), we look at each provided option to determine which represents a valid, practical step.
First, consider the original equation:
[tex]\[
\sqrt[4]{2 x-8} + \sqrt[4]{2 x+8} = 0
\][/tex]
Here, \(\sqrt[4]{y}\) denotes the fourth root of \(y\), or \(y^{1/4}\).
Next, we will analyze each of the options given:
1. Option 1:
[tex]\[
(\sqrt[4]{2 x-8})^3 = -(\sqrt[4]{2 x+8})^3
\][/tex]
Writing this in exponent form:
[tex]\[
(2 x - 8)^{3/4} = - (2 x + 8)^{3/4}
\][/tex]
This is not a valid step because raising both sides to the third power does not simplify the root in a manner consistent with our original problem, and the right side introduces unnecessary complexity.
2. Option 2:
[tex]\[
(\sqrt[4]{2 x-8})^3 = (-\sqrt[4]{2 x+8})^3
\][/tex]
In exponent form:
[tex]\[
(2 x - 8)^{3/4} = [-(2 x + 8)^{1/4}]^3
\][/tex]
This option simplifies similarly as option 1 and is not a valid step. The right-hand side adds unnecessary complexity by involving a negative sign inside a higher power.
3. Option 3:
[tex]\[
(\sqrt[4]{2 x-8})^4 = -(\sqrt[4]{2 x+8})^4
\][/tex]
In exponent form:
[tex]\[
(2 x - 8)^{4/4} = - (2 x + 8)^{4/4}
\][/tex]
Simplifying each side:
[tex]\[
2 x - 8 = - (2 x + 8)
\][/tex]
This results in a contradiction since it implies:
[tex]\[
2x - 8 = -2x - 8
\][/tex]
[tex]\[
4x = 0
\][/tex]
[tex]\[
x = 0
\][/tex]
Substitution back into the original equation does not confirm practical elimination of the root terms.
4. Option 4:
[tex]\[
(\sqrt[4]{2 x-8})^4 = (-\sqrt[4]{2 x+8})^4
\][/tex]
In exponent form:
[tex]\[
(2 x - 8)^{4/4} = [-(2 x + 8)^{1/4}]^4
\][/tex]
Simplifying each side:
[tex]\[
2 x - 8 = (-(2 x + 8))^4
\][/tex]
[tex]\[
2 x - 8 = (2 x + 8)
\][/tex]
It means:
[tex]\[
2x - 8 = -[2x + 8]
\][/tex]
[tex]\[
x = 0
\][/tex]
Solving which validates \(2x - 8 = -(2 x + 8 )\),
Therefore, the correct, practical step in solving the given equation is represented by:
[tex]\[
(\sqrt[4]{2 x-8})^4 = (-\sqrt[4]{2 x+8})^4
\][/tex]
Thus, Option 4 is correct.
