Question - Finding the Value of n

Given \( (x^n)^3 = \frac{x^{18}}{x^{-6}} \), we can start by simplifying the right side:

First, rewrite \( x^{-6} \) as \( \frac{1}{x^6} \), so we have:

\( \frac{x^{18}}{x^{-6}} = x^{18} \cdot x^{6} = x^{18 + 6} = x^{24} \)

Now we have:

\( (x^n)^3 = x^{24} \)

Using the property of exponents, we get:

\( x^{3n} = x^{24} \)

Since the bases are the same, set the exponents equal:

\( 3n = 24 \)

Now, solving for \( n \):

\( n = \frac{24}{3} = 8 \)

Thus, the value of \( n \) is \( 8 \).