To simplify the expression [tex]\(\left(125 p^{18}\right)^{-\frac{2}{3}}\)[/tex], we follow several steps that utilize the laws of exponents.
1. Simplify the coefficient part: [tex]\(125^{-\frac{2}{3}}\)[/tex]
2. Simplify the exponent part: [tex]\((p^{18})^{-\frac{2}{3}}\)[/tex]
### Step 1: Simplify the coefficient
First, let's simplify the coefficient [tex]\(125^{-\frac{2}{3}}\)[/tex]:
125 can be written as [tex]\(5^3\)[/tex]:
[tex]\[ 125 = 5^3 \][/tex]
Now, substituting this back into our expression:
[tex]\[ \left(5^3\right)^{-\frac{2}{3}} \][/tex]
Using the property of exponents [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[ 5^{3 \cdot -\frac{2}{3}} \][/tex]
Simplify the exponent:
[tex]\[ 5^{-2} \][/tex]
This can be further simplified since [tex]\(5^{-2} = \frac{1}{5^2}\)[/tex]:
[tex]\[ \frac{1}{5^2} = \frac{1}{25} \][/tex]
Therefore:
[tex]\[ 125^{-\frac{2}{3}} = \frac{1}{25} \][/tex]
### Step 2: Simplify the exponent part
Now, let's simplify [tex]\((p^{18})^{-\frac{2}{3}}\)[/tex]:
Using the property of exponents [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[ p^{18 \cdot -\frac{2}{3}} \][/tex]
Simplify the exponent:
[tex]\[ p^{-12} \][/tex]
### Combine the simplified parts
Putting it all together, we get:
[tex]\[ \frac{1}{25} \cdot p^{-12} \][/tex]
This can be written as:
[tex]\[ \frac{1}{25 p^{12}} \][/tex]
### Correct Answer
Therefore, the simplified expression [tex]\(\left(125 p^{18}\right)^{-\frac{2}{3}}\)[/tex] is:
[tex]\[ \frac{1}{25 p^{12}} \][/tex]
So, the correct answer is:
[tex]\[ \boxed{D. \frac{1}{25 p^{12}}} \][/tex]
