To solve for [tex]\( n \)[/tex] given the two equations:
[tex]\[
4 \sqrt[3]{y^2} = \sqrt[6]{x}
\][/tex]
and
[tex]\[
y = \sqrt[n]{x},
\][/tex]
we can follow these steps:
1. Express the equations in terms of exponents:
The first equation [tex]\( 4 \sqrt[3]{y^2} = \sqrt[6]{x} \)[/tex] can be rewritten using exponents. Recall that [tex]\(\sqrt[3]{y^2}\)[/tex] is [tex]\( (y^2)^{1/3} \)[/tex] and [tex]\(\sqrt[6]{x}\)[/tex] is [tex]\( x^{1/6} \)[/tex]:
[tex]\[
4 (y^2)^{1/3} = x^{1/6}
\][/tex]
Simplify the left-hand side:
[tex]\[
4 y^{2/3} = x^{1/6}
\][/tex]
2. Substitute [tex]\( y \)[/tex] from the second equation into the first equation:
From the second equation, [tex]\( y = \sqrt[n]{x} \)[/tex], you can write [tex]\( y \)[/tex] as:
[tex]\[
y = x^{1/n}
\][/tex]
Substitute [tex]\( y = x^{1/n} \)[/tex] into the first equation:
[tex]\[
4 (x^{1/n})^{2/3} = x^{1/6}
\][/tex]
Simplify the left-hand side by multiplying the exponents:
[tex]\[
4 x^{(1/n) \cdot (2/3)} = x^{1/6}
\][/tex]
[tex]\[
4 x^{2/(3n)} = x^{1/6}
\][/tex]
3. Equate the exponents since the bases are the same:
Both sides of the equation have the base [tex]\( x \)[/tex] (assuming [tex]\( x \neq 0 \)[/tex]). Therefore, we can set the exponents equal to each other:
[tex]\[
\frac{2}{3n} = \frac{1}{6}
\][/tex]
4. Solve for [tex]\( n \)[/tex]:
Cross-multiply to solve for [tex]\( n \)[/tex]:
[tex]\[
2 \cdot 6 = 1 \cdot 3n
\][/tex]
[tex]\[
12 = 3n
\][/tex]
Divide both sides by 3:
[tex]\[
n = 4
\][/tex]
Thus, the value of [tex]\( n \)[/tex] is:
[tex]\[
n = 4
\][/tex]
