Question

which term is a perfect square of the root $3x^4$?
$6x^8$
$6x^{16}$
$9x^8$
$9x^{16}$

Explanation:

Step1: Recall the square formula

To find the perfect square of a term \((ab)^n\), we use the formula \((ab)^n = a^n b^n\) and \((a^m)^n=a^{mn}\). For the term \(3x^4\), we want to find \((3x^4)^2\).

Step2: Square the coefficient and the variable separately

First, square the coefficient \(3\): \(3^2 = 9\). Then, square the variable part \(x^4\) using the power - of - a - power rule \((x^m)^n=x^{mn}\). Here, \(m = 4\) and \(n = 2\), so \((x^4)^2=x^{4\times2}=x^8\).

Step3: Combine the results

Combining the squared coefficient and the squared variable part, we get \((3x^4)^2=3^2\times(x^4)^2 = 9x^8\).

Answer:

\(9x^8\) (the option corresponding to \(9x^8\) among the given options)