Here are the answers to your questions:
Question 1: Which factors of the base would help simplify the expression \( 16^{14} \)?
- Response: \( 16 = 222*2 \)
Question 2: Given that the cube root of 27 is 3, what is \( 27^{43} \)?
- Response: \( 108 \) (This is incorrect; the correct answer is \( 27^{43} = (3^3)^{43} = 3^{129} \). In the options given, it might be a typo since it's unclear what the question actually seeks. However, it represents a large exponential instead.)
Question 3: What is the correct simplification of \( 25^{32} \)?
- Response: \( 625 \) (Since \( 25 = 5^2 \), \( 25^{32} = (5^2)^{32} = 5^{64} = 625 \))
Question 4: Given that the meaning of the rational exponent can be connected to the meaning of a root, how can you rewrite \( 3^{5-\frac{1}{2}} \) using a rational exponent?
- Response: \( 3^{5} \) (This response is incorrect; the correct way is to combine the exponent: \( 3^{5-\frac{1}{2}} = 3^{4.5} = 3^{\frac{9}{2}} \) but check which was meant by the phrasing.)
Question 5: How can you rewrite \( 8^{\frac{1}{2}} \)?
- Response: \( 8^{-\frac{1}{2}} \) is often interpreted as the square root of 8, so the answer is square root of 8
If you require further clarity or more questions, feel free to ask!