Let's simplify the expression \(\left(\frac{3}{x}\right)^4\) step by step.
1. Initial Expression:
[tex]\[
\left(\frac{3}{x}\right)^4
\][/tex]
2. Applying the Exponent:
When you raise a fraction to a power, you apply the exponent to both the numerator and the denominator. So,
[tex]\[
\left(\frac{3}{x}\right)^4 = \frac{3^4}{x^4}
\][/tex]
3. Calculating the Exponents:
Calculate \(3^4\):
[tex]\[
3^4 = 3 \times 3 \times 3 \times 3 = 81
\][/tex]
The \(x^4\) term remains as it is:
[tex]\[
x^4
\][/tex]
4. Final Simplified Expression:
Combine the results from the numerator and the denominator:
[tex]\[
\frac{3^4}{x^4} = \frac{81}{x^4}
\][/tex]
So, the simplified expression is:
[tex]\[
\frac{81}{x^4}
\][/tex]
We can write it as \(\frac{81}{x}\). Therefore, the answer to what should be written in the numerator is 81.
[tex]\[
\frac{81}{x^4}
\][/tex]
