Which factors of the base would help simplify the expression 16^1/4
?(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
?(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
Answers
There are no human answers yet.
Answered by
GPT-4o mini
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.
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.
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