Let's tackle each question step by step:
Question 1: Who has the correct value for simplifying \(100023\)?
The value \(100023\) can be interpreted in terms of its factors of 10 and represents \(1000^2\) as it has a clear structure of 10 to the power of 6 (since \(1000000 = 10^6\)). Hence:
- Charles's interpretation of \(100023 = 10\) is incorrect.
- Zayeer's interpretation of \(100023 = 1000\) is incorrect. He confuses bases.
- Kali's interpretation of \(100023 = 100\) is also incorrect.
Thus, none of them is correct.
Question 2: Simplify \(24335\).
To simplify this number, we factor it: \[ 24335 = 5 \times 4867 \] The number cannot be simplified further easily without a specific request for factors or primes.
So the answer remains as \(24335\).
Question 3: Write the radical form of the expression \(2514\).
This means expressing it as: \[ 2514 = 25^{7} = (5^2)^{14} = 5^{28} \] The radical form is not explicitly to show this. However, it depends on using rational roots as well, which were not clearly asked for.
Thus, \(2514 = \sqrt[14]{25^2}\) makes sense in this interpretation: \[ = \sqrt[14]{25^{14}} = 25 \text{ (since the exponent and the root match)} \]
Question 4: What is the simplest form of \(2512\)?
\[ 25^{12} = 5^{24} \text{ or } \sqrt{5^{24}} = 5^{12} \] The simplest form is \(5^{12}\).
Question 5: Rewrite the expressions \(2712 \cdot 2714\).
Using the property of exponents: \[ a^m \cdot a^n = a^{m+n} \] So we get: \[ 27^{12+14} = 27^{26} \]
Question 6: What is an equivalent expression for \(- (b^{14})^{12}\)?
Using power of a power: \[
- (b^{14})^{12} = - b^{168} \] None of the responses given fits perfectly, but \(- b^{168}\) would be correct.
Question 7: Find the value of \(x\) in \(-x^{32} = -27\).
Removing negatives from both sides gives: \[ x^{32} = 27 \implies x = 27^{1/32} \]
Question 8: Solve for \(x\) in \(27x^{34} - 1 = 26\).
Adding to both sides: \[ 27x^{34} = 27 \implies x^{34} = 1 \implies x = 1 \text{ or } -1 \]
Question 9: Using exponent properties, which of the following expressions is equivalent to \((2^{-\frac{1}{2}})^{3} \cdot 2^{-1}\)?
First calculate: \[ (2^{-\frac{1}{2}})^{3} = 2^{-\frac{3}{2}} \]
Then multiply: \[ = 2^{-\frac{3}{2} - 1} = 2^{- \frac{5}{2}} \text{ or } \frac{1}{\sqrt{2^5}} = 2^{-\frac{5}{2}} \]
Question 10: What is the equivalent expression to \(216\sqrt{32713}\)?
216 becomes \(6^3\), and \(\sqrt{32713}\) doesn't simplify easily without values.
Question 11: The equation \((8\sqrt{4})^{-2} \cdot (8\sqrt{})^{16} = 8^{-48}\) is false.
Thus, your explanation text must point at exponents.
Question 12: Simplify the expression \(7^{-4} \cdot 7349\).
Using exponent properties, this results in a fraction form without negatives left in the final form.
If you have any question or need further details on any specific solution, feel free to ask!