First, let's simplify the number 912912.
To do this, we can look for the prime factors of the number:
- 912912 can be factored as follows:
- First, we can notice that it is even, hence divisible by 2:
\( 912912 \div 2 = 456456 \) \( 456456 \div 2 = 228228 \) \( 228228 \div 2 = 114114 \) \( 114114 \div 2 = 57057 \)
Now, 57057 is no longer even. Next, we can check for divisibility by 3:
- The sum of the digits of 57057 (5 + 7 + 0 + 5 + 7 = 24) is divisible by 3.
\( 57057 \div 3 = 19019 \)
Next, let's check 19019 for further factorization:
- Checking for 7:
\( 19019 \div 7 = 2717 \)
We can keep factorizing:
- Checking for further factors of 2717:
\( 2717 \div 11 = 247 \)
Finally:
- 247 can be factored further:
\( 247 \div 13 = 19 \)
So the prime factorization of 912912 is: \[ 912912 = 2^4 \times 3^1 \times 7^1 \times 11^1 \times 13^1 \times 19^1 \] We can see that this is simplified fully in terms of prime factors.
For Question 2, let's rewrite and simplify \( \sqrt{81} \cdot 4^{14} \).
-
The square root of 81 is 9:
- \( \sqrt{81} = 9 \)
-
For \( 4^{14} \):
- We can also rewrite \( 4 \) as \( 2^2 \), so:
- \( 4^{14} = (2^2)^{14} = 2^{28} \)
Thus, we can express the whole expression: \[ \sqrt{81} \cdot 4^{14} = 9 \cdot 2^{28} \]
To find a rational exponent for \( 81^{1/2} \), we can denote it as: \[ 81^{1/2} = (3^4)^{1/2} = 3^{4/2} = 3^2 = 9 \]
Thus, we can express:
- The rational number that could be used as an exponent is 1/2,
- The expression simplifies to \(9 \cdot 2^{28}\).