To simplify the expression \( p^{-7} + p^{-12} \), we can begin by rewriting the terms with positive exponents. The negative exponent rule states that \( p^{-n} = \frac{1}{p^n} \).
So, we can rewrite the expression as follows:
\[ p^{-7} + p^{-12} = \frac{1}{p^7} + \frac{1}{p^{12}} \]
Next, we need to add these two fractions. To do this, we need a common denominator. The least common denominator (LCD) of \( p^7 \) and \( p^{12} \) is \( p^{12} \).
Now, we rewrite each term with the common denominator:
\[ \frac{1}{p^7} = \frac{p^5}{p^{12}} \quad \text{(since } \frac{1}{p^7} \cdot \frac{p^5}{p^5} = \frac{p^5}{p^{12}} \text{)} \]
Now, we can express the original expression with the common denominator:
\[ \frac{p^5}{p^{12}} + \frac{1}{p^{12}} = \frac{p^5 + 1}{p^{12}} \]
Thus, the simplified form of the expression \( p^{-7} + p^{-12} \) is:
\[ \frac{p^5 + 1}{p^{12}} \]
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