Let's go through each of the questions systematically.
Question 1:
Who has the correct value among Charles, Zayeer, and Kali regarding the simplification of \(100023\)?
- Charles states \(1000 = 10^3\) and \(100023 = 10^1\), which is incorrect.
- Zayeer states \(1000 = 10^3\) and simplistically calculates \(100023 = 10 \cdot 10 \cdot 10 = 1000\), which is incorrect in context.
- Kali deduces \(1000 = 10^3\) and simplifies \(100023\) to be \(10 \cdot 10 = 100\), which again is incorrect.
Correct Value: None of them.
Question 2:
Simplify \(24335\)
Since \(24335\) is not conducive to simplification as the number itself, it's likely meant to be interpreted as \(2^4 \cdot 3^3 \cdot 5^5\). Thus,
Answer: \(24335\) simplified is \(2^4 \cdot 3^3 \cdot 5^5\) or you would typically state it as is.
Question 3:
Write the radical form of the expression \(2514\):
The radical form of \(25^{14}\) can be written as: \[ (5^2)^{14} = 5^{28} \] But the simplest radical form of this expression does not apply if treated directly.
Answer: \( (5^{28}) \)
Question 4:
What is the simplest form of \(2512\)?
This means \((25^{12})\), and since \(25 = 5^2\): \[ (5^2)^{12} = 5^{24} \] The square root of \(25\) reduces to \(5^{12}\), and thus.
Answer: \(5^{12}\)
Question 5:
Rewrite the following expression \(2712 \cdot 2714\):
Using properties of exponents, add the exponents: \[ 27^{12} \cdot 27^{14} = 27^{12+14} = 27^{26} \] So, Answer: \(27^{26}\)
Question 6:
What is an equivalent expression for \(- (b^{14})^{12}\)?
Using the power of a power property: \[ -(b^{14})^{12} = -b^{168} = -b^{(-168)} \]
Answer: \(-b^{168}\)
Question 7:
Find the value of \(x\) in \(−x^{32} = −27\):
Removing the negative sign via division: \[ x^{32} = 27 \quad \Rightarrow \quad x = 27^{1/32} \] Since \(27 = 3^3\): \[ x = 3^{3/32} \]
Question 8:
Solve for \(x\) in \(27x^{34} - 1 = 26\)
Rearranging gives: \[ 27x^{34} = 27 \quad \Rightarrow \quad x^{34} = 1 \ x = 1^{1/34} = 1 \]
Question 9:
Using exponent properties, which of the following expressions is equivalent to \((2^{-1/2})^3 \cdot 2^{-12}\):
Expanding: \[ (2^{-1/2})^3 = 2^{-3/2} \] Adding the exponents: \[ 2^{-3/2} \cdot 2^{-12} = 2^{-3/2 - 12} = 2^{-27/2} \]
Question 10:
What is the simplified form of the expression \(2^{16} \sqrt{3^{27} 17^{13}}\)?
Simplifying \( \sqrt{3^{27} 17^{13}} \) gives \(3^{27/2} \cdot 17^{13/2}\) and overall we have: \[ 2^{16} \cdot 3^{13.5} \cdot 17^{6.5} \]
Question 11:
Explanation of the mathematics in \( (8 - \sqrt{4})^{-24} \cdot (8 - \sqrt{})^{16} = 8^{-48}\)
The equation isn’t justified. Needs checking of exponents combined/mistake in simplification.
Question 12:
Simplify the expression \(7^{-4}\cdot 7349\sqrt{}\):
Use properties of exponentials: \[ \text{That would require direct calculation. } \] Since answer formatting isn't visible, it would be approached individually.
These are the refined approaches for each question. Let me know if you need any clarifications!