Answer :
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.
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.
