To simplify the expression \((3m^2n^9p^9)^2\), we apply the power of a product property. This property states that \((ab)^n = a^n b^n\).
First, we will square each part of the expression:
\[ (3m^2n^9p^9)^2 = 3^2(m^2)^2(n^9)^2(p^9)^2 \]
Now we calculate each part:
- \(3^2 = 9\)
- \((m^2)^2 = m^{2 \times 2} = m^4\)
- \((n^9)^2 = n^{9 \times 2} = n^{18}\)
- \((p^9)^2 = p^{9 \times 2} = p^{18}\)
Putting it all together gives:
\[ 9m^4n^{18}p^{18} \]
Thus, the simplified expression is indeed \(9m^4n^{18}p^{18}\).
The statement that \((3m^2n^9p^9)^2\) is equivalent to \(9m^4n^{18}p^{18}\) is True.