To find the value of [tex]\( n \)[/tex] given the equations [tex]\(\sqrt[3]{y^2} = \sqrt[6]{x}\)[/tex] and [tex]\( y = \sqrt[n]{x} \)[/tex], we will solve the system step-by-step.
### Step 1: Rewrite the first equation
The first equation is:
[tex]\[
\sqrt[3]{y^2} = \sqrt[6]{x}
\][/tex]
This can be rewritten using exponents:
[tex]\[
(y^2)^{1/3} = x^{1/6}
\][/tex]
Simplify the left side:
[tex]\[
y^{2/3} = x^{1/6}
\][/tex]
### Step 2: Rewrite the second equation
The second equation is:
[tex]\[
y = \sqrt[n]{x}
\][/tex]
This can also be rewritten using exponents:
[tex]\[
y = x^{1/n}
\][/tex]
### Step 3: Substitute [tex]\( y \)[/tex] in the first equation
Substitute [tex]\( y = x^{1/n} \)[/tex] into the first equation [tex]\( y^{2/3} = x^{1/6} \)[/tex]:
[tex]\[
(x^{1/n})^{2/3} = x^{1/6}
\][/tex]
Simplify the left side by multiplying the exponents:
[tex]\[
x^{(1/n) \cdot (2/3)} = x^{1/6}
\][/tex]
This results in:
[tex]\[
x^{2/(3n)} = x^{1/6}
\][/tex]
### Step 4: Equate the exponents
Since the bases are the same, the exponents must be equal:
[tex]\[
\frac{2}{3n} = \frac{1}{6}
\][/tex]
### Step 5: Solve for [tex]\( n \)[/tex]
To solve for [tex]\( n \)[/tex], cross-multiply:
[tex]\[
2 \cdot 6 = 1 \cdot 3n
\][/tex]
Simplify:
[tex]\[
12 = 3n
\][/tex]
Divide both sides by 3:
[tex]\[
n = \frac{12}{3}
\][/tex]
[tex]\[
n = 4
\][/tex]
### Conclusion
The value of [tex]\( n \)[/tex] is:
[tex]\[
\boxed{4}
\][/tex]
