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Aplique propiedades de potencias según corresponda, exprese su resultado en notación potencia

The expression given is:

\[
\frac{35^{11} \cdot 49^4 \cdot (-12)^{-31}}{10^{12} \cdot 6^{30} \cdot (-14)^{20}}
\]

We will simplify the expression using the properties of exponents.

### Step 1: Express numbers with common bases

First, note that \(49\) and \(35\) can be expressed as powers of prime numbers:
- \(49 = 7^2\), so \(49^4 = (7^2)^4 = 7^8\).

This changes the original expression to:

\[
\frac{35^{11} \cdot 7^8 \cdot (-12)^{-31}}{10^{12} \cdot 6^{30} \cdot (-14)^{20}}
\]

### Step 2: Rewrite products as prime factors

- \(35 = 5 \cdot 7\), so \(35^{11} = (5 \cdot 7)^{11} = 5^{11} \cdot 7^{11}\).
- \(10 = 2 \cdot 5\), so \(10^{12} = (2 \cdot 5)^{12} = 2^{12} \cdot 5^{12}\).
- \(6 = 2 \cdot 3\), so \(6^{30} = (2 \cdot 3)^{30} = 2^{30} \cdot 3^{30}\).
- \(-12 = (-1) \cdot 2^2 \cdot 3\), so \((-12)^{-31} = (-1)^{-31} \cdot 2^{-62} \cdot 3^{-31}\).
- \(-14 = (-1) \cdot 2 \cdot 7\), so \((-14)^{20} = (-1)^{20} \cdot 2^{20} \cdot 7^{20}\).

Substitute these into the expression:

\[
\frac{(5^{11} \cdot 7^{11}) \cdot 7^8 \cdot (-1)^{-31} \cdot 2^{-62} \cdot 3^{-31}}{(2^{12} \cdot 5^{12}) \cdot (2^{30} \cdot 3^{30}) \cdot (-1)^{20} \cdot 2^{20} \cdot 7^{20}}
\]

### Step 3: Combine like terms

Now, simplify the terms with the same bases:

- Base 5: \( \frac{5^{11}}{5^{12}} = 5^{11-12} = 5^{-1}\)
- Base 7: \( \frac{7^{11} \cdot 7^8}{7^{20}} = \frac{7^{19}}{7^{20}} = 7^{-1}\)
- Base \(-1\): \( (-1)^{-31} \cdot (-1)^{-20} = (-1)^{-31 + 20} = (-1)^{-11}\)
- Base 2: \( \frac{2^{-62}}{2^{12 + 30 + 20}} = \frac{2^{-62}}{2^{62}} = 2^{-62 + 62} = 2^0 = 1 \) (cancels out)
- Base 3: \( \frac{3^{-31}}{3^{30}} = 3^{-31 - 30} = 3^{-61}\)

### Final Expression

The final simplified result is:

\[
\frac{5^{-1} \cdot 7^{-1} \cdot (-1)^{-11} \cdot 3^{-61}}{1} = \frac{(-1)^{-11}}{5 \cdot 7 \cdot 3^{61}}
\]

Since \((-1)^{-11} = -1\), the final answer is:

\[
-\frac{1}{5 \cdot 7 \cdot 3^{61}} = -\frac{1}{35 \cdot 3^{61}}
\]

Thus, the result in power notation is:

\[
-\frac{1}{35 \cdot 3^{61}}
\]

---

Would you like further explanation on any part of the process?

Here are some related questions to explore further:
1. How do negative exponents affect the terms in an expression?
2. Can you explain why \((-1)^{-11} = -1\)?
3. How would the result change if the exponents of \(-12\) and \(-14\) were different?
4. What is the impact of zero exponents in such expressions?
5. How can prime factorization simplify complex expressions like this?

**Tip**: When simplifying expressions, always try to factor numbers into primes, as this makes it easier to apply exponent rules across different terms.

This page explains how to simplify a complex fraction involving exponentiation, such as \( \frac{35^{11} \cdot 49^4 \cdot (-12)^{-31}}{10^{12} \cdot 6^{30} \cdot (-14)^{20}} \), using properties of exponents and prime factorization. Suitable for students in Grades 10-12.

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