To solve the given system of equations:
1. [tex]\(4 \sqrt[3]{y^2} = \sqrt[6]{x}\)[/tex]
2. [tex]\(y = \sqrt[4]{x}\)[/tex]
we will follow a step-by-step approach.
### Step 1: Solve for [tex]\( y \)[/tex] using the second equation
The second equation is:
[tex]\[ y = \sqrt[4]{x} \][/tex]
This can also be written in exponential form as:
[tex]\[ y = x^{1/4} \][/tex]
### Step 2: Substitute [tex]\( y = x^{1/4} \)[/tex] into the first equation
The first equation given is:
[tex]\[ 4 \sqrt[3]{y^2} = \sqrt[6]{x} \][/tex]
Substituting [tex]\( y = x^{1/4} \)[/tex] into the first equation gives:
[tex]\[ 4 \sqrt[3]{(x^{1/4})^2} = \sqrt[6]{x} \][/tex]
[tex]\[ 4 \sqrt[3]{x^{2/4}} = \sqrt[6]{x} \][/tex]
[tex]\[ 4 \sqrt[3]{x^{1/2}} = \sqrt[6]{x} \][/tex]
### Step 3: Express the radicals with exponents
Rewrite both sides using exponential notation:
[tex]\[ 4 (x^{1/2})^{1/3} = x^{1/6} \][/tex]
[tex]\[ 4 x^{(1/2) \cdot (1/3)} = x^{1/6} \][/tex]
[tex]\[ 4 x^{1/6} = x^{1/6} \][/tex]
### Step 4: Solve the equation
For the equation to hold true:
[tex]\[ 4 x^{1/6} = x^{1/6} \][/tex]
This implies:
[tex]\[ 4 = 1 \times x^{1/6} \][/tex]
[tex]\[ 4 = x^{1/6} \][/tex]
### Step 5: Solve for [tex]\( x \)[/tex]
Raise both sides to the 6th power to solve for [tex]\( x \)[/tex]:
[tex]\[ (4)^{6} = x \][/tex]
[tex]\[ x = 4^6 \][/tex]
[tex]\[ x = 4096 \][/tex]
Thus, the value of [tex]\( x \)[/tex] is 4096.
### Step 6: Verify the corresponding [tex]\( y \)[/tex]
Using [tex]\( y = x^{1/4} \)[/tex]:
[tex]\[ y = 4096^{1/4} \][/tex]
[tex]\[ y = 6 = 16 \][/tex]
### Step 7: Solve for the value of [tex]\( n \)[/tex]
Here, the value of [tex]\( n \)[/tex] is the value of [tex]\( x \)[/tex]. Therefore,
[tex]\[
\boxed{0}
\][/tex]
