Which factors of the base would help simplify the expression 16^1/4

Question

?(1 point)
Responses

16=4⋅2⋅2
16 equals 4 times 2 times 2

16=4⋅4
16 equals 4 times 4

16=8⋅8
16 equals 8 times 8

16=2⋅2⋅2⋅2

Given that the cube root of 27 is 3, what is 27^4/3
?(1 point)
Responses

4
4

12
12

81
81

108

What is the correct simplification of 25^3/2
?(1 point)
Responses

37.5
37.5

625
625

75
75

125

Given that the meaning of a rational exponent can be connected to the meaning of a root, how can you rewrite 5–√3
using a rational exponent?(1 point)
Responses

51
5 superscript 1 baseline

35
3 superscript 5 baseline

53
5 cubed

513
5 superscript Start Fraction 1 over 3 End Fraction baseline

How can you rewrite 8^1/2
using a root?(1 point)
Responses

12−−√8
root index 8 Start Root Start Fraction 1 over 2 End Fraction End Root

8–√1
root index 1 Start Root 8 End Root

8–√12
root index Start Fraction 1 over 2 End Fraction Start Root 8 End Root

8–√
Start Root 8 End Root

Answer 1

Let's address each part of your question step by step.

1. Which factors of the base would help simplify the expression \( 16^{1/4} \)?

   • The best factorization for simplification is:
     • 16 = 4 ⋅ 4
     • This allows us to express \( 16^{1/4} = (4^2)^{1/4} = 4^{2 \cdot (1/4)} = 4^{1/2} = \sqrt{4} = 2 \).
   • Alternatively, we could also express 16 as:
     • 16 = 2 ⋅ 2 ⋅ 2 ⋅ 2 (which indicates that \( 16 = 2^4 \)).
   • But for clarity, the most straightforward simplification comes from using \( 16 = 4 \cdot 4 \).

2. Given that the cube root of 27 is 3, what is \( 27^{4/3} \)?

   • Since \( 27^{1/3} = 3 \), we can rewrite \( 27^{4/3} \) as:
     • \( (27^{1/3})^4 = 3^4 = 81 \).
   • So the answer is 81.

3. What is the correct simplification of \( 25^{3/2} \)?

   • We can express \( 25^{3/2} \) as:
     • \( (25^{1/2})^3 = (5)^3 = 125 \).
   • Thus, the answer is 125.

4. Given that the meaning of a rational exponent can be connected to the meaning of a root, how can you rewrite \( 5^{-\sqrt{3}} \) using a rational exponent?

   • The notation \( 5^{-\sqrt{3}} \) is already in the form of rational exponent but if you want it in the form of \( 5^{-3^{1/2}} \), that works also.
   • The correct response would be:
     • \( 5^{-\frac{1}{3}} \) (which is not explicitly listed here).

5. How can you rewrite \( 8^{1/2} \) using a root?

   • The expression \( 8^{1/2} \) represents the square root of 8, which can be simplified to:
     • \( \sqrt{8} \) or written out as \( 8^{1/2} = \sqrt{8} \).
   • The best option that closely aligns is:
     • \( \sqrt{8} \) which can be termed as \( 8^{1/2} \).

If you look for specific answers based on your listed responses for the last few parts, here's a summary of the answers:

1. Factors that help simplify \( 16^{1/4} \): \(16 = 4 \cdot 4\)
2. \( 27^{4/3} = 81 \)
3. \( 25^{3/2} = 125 \)
4. \( 5^{-\sqrt{3}} = 5^{-3^{1/2}} \)
5. Rewrite \( 8^{1/2} = \sqrt{8} \) or \( \sqrt{8} \) is a direct connection.