Asked by adaleta
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=8⋅8
16 equals 8 times 8
16=4⋅4
16 equals 4 times 4
16=2⋅2⋅2⋅2
16 equals 2 times 2 times 2 times 2
Responses
16=4⋅2⋅2
16 equals 4 times 2 times 2
16=8⋅8
16 equals 8 times 8
16=4⋅4
16 equals 4 times 4
16=2⋅2⋅2⋅2
16 equals 2 times 2 times 2 times 2
Answers
Answered by
adaleta
Given that the cube root of 27 is 3, what is 27 4/3
?(1 point)
Responses
12
12
81
81
108
108
4
?(1 point)
Responses
12
12
81
81
108
108
4
Answered by
adaleta
What is the correct simplification of 25 3/2?(1 point)
Responses
37.5
37.5
125
125
75
75
625
Responses
37.5
37.5
125
125
75
75
625
Answered by
adaleta
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
513
5 superscript Start Fraction 1 over 3 End Fraction baseline
53
5 cubed
51
5 superscript 1 baseline
35
Responses
513
5 superscript Start Fraction 1 over 3 End Fraction baseline
53
5 cubed
51
5 superscript 1 baseline
35
Answered by
adaleta
so what is the correct answer
Answered by
adaleta
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–√1/2
root index Start Fraction 1 over 2 End Fraction Start Root 8 End Root
8–√1
root index 1 Start Root 8 End Root
8–√
Answered by
GPT-4o mini
To simplify the expression \(16 \frac{1}{4}\), we first convert it into an improper fraction.
1. **Convert \(16 \frac{1}{4}\) to an improper fraction**:
\[
16 \frac{1}{4} = 16 + \frac{1}{4} = \frac{64}{4} + \frac{1}{4} = \frac{65}{4}
\]
Now, regarding the factors of the base \(16\) that would help in simplification, we observe the following:
- \(16 = 2^4\)
- \(16 = 4 \times 4\)
- \(16 = 8 \times 2\)
Among the options given, the most useful factorization for simplifications that involve \(2\) (often helpful for calculations and further simplifications) would be:
**16 = 2⋅2⋅2⋅2** (16 equals 2 times 2 times 2 times 2)
This representation allows us to easily manipulate and simplify expressions involving powers of \(2\).
1. **Convert \(16 \frac{1}{4}\) to an improper fraction**:
\[
16 \frac{1}{4} = 16 + \frac{1}{4} = \frac{64}{4} + \frac{1}{4} = \frac{65}{4}
\]
Now, regarding the factors of the base \(16\) that would help in simplification, we observe the following:
- \(16 = 2^4\)
- \(16 = 4 \times 4\)
- \(16 = 8 \times 2\)
Among the options given, the most useful factorization for simplifications that involve \(2\) (often helpful for calculations and further simplifications) would be:
**16 = 2⋅2⋅2⋅2** (16 equals 2 times 2 times 2 times 2)
This representation allows us to easily manipulate and simplify expressions involving powers of \(2\).
Answered by
GPT-4o mini
To find \(27^{\frac{4}{3}}\), we can start by using the property of exponents that states:
\[
a^{\frac{m}{n}} = \left( a^{m} \right)^{\frac{1}{n}}
\]
In this case, we can rewrite \(27^{\frac{4}{3}}\) as:
\[
27^{\frac{4}{3}} = \left(27^{\frac{1}{3}}\right)^{4}
\]
We know that the cube root of \(27\) is \(3\):
\[
27^{\frac{1}{3}} = 3
\]
Now we can substitute this value back into the expression:
\[
\left(27^{\frac{1}{3}}\right)^{4} = (3)^{4} = 3 \times 3 \times 3 \times 3 = 81
\]
Thus, \(27^{\frac{4}{3}} = 81\).
So, the answer is:
**81**
\[
a^{\frac{m}{n}} = \left( a^{m} \right)^{\frac{1}{n}}
\]
In this case, we can rewrite \(27^{\frac{4}{3}}\) as:
\[
27^{\frac{4}{3}} = \left(27^{\frac{1}{3}}\right)^{4}
\]
We know that the cube root of \(27\) is \(3\):
\[
27^{\frac{1}{3}} = 3
\]
Now we can substitute this value back into the expression:
\[
\left(27^{\frac{1}{3}}\right)^{4} = (3)^{4} = 3 \times 3 \times 3 \times 3 = 81
\]
Thus, \(27^{\frac{4}{3}} = 81\).
So, the answer is:
**81**
Answered by
GPT-4o mini
To simplify \(25 \frac{3}{2}\), we first convert it into an improper fraction.
1. **Convert** \(25 \frac{3}{2}\) **to an improper fraction**:
\[
25 \frac{3}{2} = 25 + \frac{3}{2}
\]
To add these, we convert \(25\) into a fraction that has the same denominator:
\[
25 = \frac{50}{2} \quad \text{(since } 25 \times 2 = 50\text{)}
\]
Now add the fractions:
\[
\frac{50}{2} + \frac{3}{2} = \frac{50 + 3}{2} = \frac{53}{2}
\]
2. **Simplifying \(25^{\frac{3}{2}}\)**:
Alternatively, if the expression means \(25^{\frac{3}{2}}\), we likely need to calculate it that way.
\[
25^{\frac{3}{2}} = \left(25^{\frac{1}{2}}\right)^{3}
\]
The square root of \(25\) is \(5\):
\[
25^{\frac{1}{2}} = 5
\]
Now raise \(5\) to the power of \(3\):
\[
5^{3} = 5 \times 5 \times 5 = 125
\]
Therefore, if you meant \(25^{\frac{3}{2}}\), the correct simplification is:
**125**.
If you were asking about \(25 \frac{3}{2}\) as an improper fraction, it results in \(\frac{53}{2}\), which is equivalent to \(26.5\) but among the provided options, \(125\) is indeed the correct answer for \(25^{\frac{3}{2}}\).
1. **Convert** \(25 \frac{3}{2}\) **to an improper fraction**:
\[
25 \frac{3}{2} = 25 + \frac{3}{2}
\]
To add these, we convert \(25\) into a fraction that has the same denominator:
\[
25 = \frac{50}{2} \quad \text{(since } 25 \times 2 = 50\text{)}
\]
Now add the fractions:
\[
\frac{50}{2} + \frac{3}{2} = \frac{50 + 3}{2} = \frac{53}{2}
\]
2. **Simplifying \(25^{\frac{3}{2}}\)**:
Alternatively, if the expression means \(25^{\frac{3}{2}}\), we likely need to calculate it that way.
\[
25^{\frac{3}{2}} = \left(25^{\frac{1}{2}}\right)^{3}
\]
The square root of \(25\) is \(5\):
\[
25^{\frac{1}{2}} = 5
\]
Now raise \(5\) to the power of \(3\):
\[
5^{3} = 5 \times 5 \times 5 = 125
\]
Therefore, if you meant \(25^{\frac{3}{2}}\), the correct simplification is:
**125**.
If you were asking about \(25 \frac{3}{2}\) as an improper fraction, it results in \(\frac{53}{2}\), which is equivalent to \(26.5\) but among the provided options, \(125\) is indeed the correct answer for \(25^{\frac{3}{2}}\).
Answered by
GPT-4o mini
To rewrite \(5 - \sqrt{3}\) using a rational exponent, we first need to clarify that \(\sqrt{3}\) can be expressed as a fractional exponent.
\(\sqrt{3}\) can be rewritten as \(3^{\frac{1}{2}}\). Therefore, the expression \(5 - \sqrt{3}\) can be written as:
\[
5 - 3^{\frac{1}{2}}
\]
However, if you're looking specifically for an expression involving a rational exponent related to \(5 - \sqrt{3}\) in a specific format (like one of the provided responses), there isn't a straightforward match among the answers given.
But if we consider rewriting just \(-\sqrt{3}\) as \(-3^{\frac{1}{2}}\), then the main focus should be on expressing it as a positive power like \(5^{1}\) minus something that can be captured in a rational exponent, it would still not align perfectly with the response options.
Given the options you provided, let’s analyze each:
1. **\(5^{\frac{1}{3}}\)**: This does not relate to \(\sqrt{3}\).
2. **\(5^3\)**: This is simply \(5\) cubed, again not related.
3. **\(5^1\)**: This equals \(5\), and doesn't relate to \(\sqrt{3}\).
4. **\(3^5\)**: This is \(3\) raised to the fifth power and doesn’t correlate either.
None of the provided responses directly express \(5 - \sqrt{3}\). However, if you need just the rational exponent form for \(\sqrt{3}\), the corresponding exponent would be \(3^{\frac{1}{2}}\).
If you meant to reflect just the square root of \(3\) with a rational exponent, your focus might be on the \(3^{\frac{1}{2}}\) part instead. But in the context of \(5 - \sqrt{3}\) as an overall expression, none of the options strictly apply.
Let me know if we can reformulate based on any further clarifications!
\(\sqrt{3}\) can be rewritten as \(3^{\frac{1}{2}}\). Therefore, the expression \(5 - \sqrt{3}\) can be written as:
\[
5 - 3^{\frac{1}{2}}
\]
However, if you're looking specifically for an expression involving a rational exponent related to \(5 - \sqrt{3}\) in a specific format (like one of the provided responses), there isn't a straightforward match among the answers given.
But if we consider rewriting just \(-\sqrt{3}\) as \(-3^{\frac{1}{2}}\), then the main focus should be on expressing it as a positive power like \(5^{1}\) minus something that can be captured in a rational exponent, it would still not align perfectly with the response options.
Given the options you provided, let’s analyze each:
1. **\(5^{\frac{1}{3}}\)**: This does not relate to \(\sqrt{3}\).
2. **\(5^3\)**: This is simply \(5\) cubed, again not related.
3. **\(5^1\)**: This equals \(5\), and doesn't relate to \(\sqrt{3}\).
4. **\(3^5\)**: This is \(3\) raised to the fifth power and doesn’t correlate either.
None of the provided responses directly express \(5 - \sqrt{3}\). However, if you need just the rational exponent form for \(\sqrt{3}\), the corresponding exponent would be \(3^{\frac{1}{2}}\).
If you meant to reflect just the square root of \(3\) with a rational exponent, your focus might be on the \(3^{\frac{1}{2}}\) part instead. But in the context of \(5 - \sqrt{3}\) as an overall expression, none of the options strictly apply.
Let me know if we can reformulate based on any further clarifications!
Answered by
GPT-4o mini
The expression \(5 - \sqrt{3}\) itself does not directly translate to any of the provided response options. However, if the intention is to express the square root of \(3\) using a rational exponent, we could rewrite \(\sqrt{3}\) as \(3^{\frac{1}{2}}\).
However, since that does not seem relevant to the provided options, it looks like you may actually be looking for the rational exponent form of just \(-\sqrt{3}\) rather than the entire expression \(5 - \sqrt{3}\). Here's a breakdown of the options based on that realization:
1. **\(5^{\frac{1}{3}}\)**: This indicates the cube root of 5, which is not related.
2. **\(5^3\)**: This means \(5\) raised to the power of \(3\), irrelevant.
3. **\(5^1\)**: This is simply \(5\), irrelevant to \(\sqrt{3}\).
4. **\(3^5\)**: This refers to \(3\) raised to the power of \(5\), again irrelevant.
Given all that, none of the response options directly express \(5 - \sqrt{3}\) or \(\sqrt{3}\) as \(3^{\frac{1}{2}}\) in the context as it was posed.
If you're asking how to express \(\sqrt{3}\) as a rational exponent, it would be \(3^{\frac{1}{2}}\).
Therefore, the answer does not directly correspond to any of the response choices you've listed, but **the most apt representation of the expression \(\sqrt{3}\) in rational exponent form is \(3^{\frac{1}{2}}\)**, if that translation is relevant.
If you'd like to specify whether you want just \(\sqrt{3}\) or the entire expression \(5 - \sqrt{3}\), that would help clarify! Otherwise, in terms of just rational exponents, that option is not listed.
However, since that does not seem relevant to the provided options, it looks like you may actually be looking for the rational exponent form of just \(-\sqrt{3}\) rather than the entire expression \(5 - \sqrt{3}\). Here's a breakdown of the options based on that realization:
1. **\(5^{\frac{1}{3}}\)**: This indicates the cube root of 5, which is not related.
2. **\(5^3\)**: This means \(5\) raised to the power of \(3\), irrelevant.
3. **\(5^1\)**: This is simply \(5\), irrelevant to \(\sqrt{3}\).
4. **\(3^5\)**: This refers to \(3\) raised to the power of \(5\), again irrelevant.
Given all that, none of the response options directly express \(5 - \sqrt{3}\) or \(\sqrt{3}\) as \(3^{\frac{1}{2}}\) in the context as it was posed.
If you're asking how to express \(\sqrt{3}\) as a rational exponent, it would be \(3^{\frac{1}{2}}\).
Therefore, the answer does not directly correspond to any of the response choices you've listed, but **the most apt representation of the expression \(\sqrt{3}\) in rational exponent form is \(3^{\frac{1}{2}}\)**, if that translation is relevant.
If you'd like to specify whether you want just \(\sqrt{3}\) or the entire expression \(5 - \sqrt{3}\), that would help clarify! Otherwise, in terms of just rational exponents, that option is not listed.
Answered by
GPT-4o mini
To rewrite \(8^{\frac{1}{2}}\) using a root, you should recognize that the exponent \(\frac{1}{2}\) indicates the square root of the base.
Thus, \(8^{\frac{1}{2}}\) can be rewritten as:
\[
\sqrt{8}
\]
Now, let's look at the provided response options to see which one correctly represents this:
1. **\(12−−√8\)**: This doesn't correctly express \(8^{\frac{1}{2}}\).
2. **\(\sqrt[2]{8}\)**: This correctly represents \(8^{\frac{1}{2}}\) as the square root of \(8\).
3. **\(8−\sqrt{\frac{1}{2}}\)**: This incorrectly alters the original expression.
4. **\(\sqrt[1]{8}\)**: This is just \(8\) since the first root is the number itself.
5. **\(8−\sqrt{}\)**: This is an incomplete expression and doesn't apply.
From the options provided, the correct way to rewrite \(8^{\frac{1}{2}}\) (interpreted as a square root) is:
**\(\sqrt{8}\)**
or, if your intended notation implies \(\sqrt[2]{8}\), that is also valid, as both refer to the square root of \(8\).
If \(\sqrt[2]{8}\) is represented in your options as **\(\sqrt[2]{8}\)** (though it wasn't formatted in your query), that would also be correct.
Thus, \(8^{\frac{1}{2}}\) can be rewritten as:
\[
\sqrt{8}
\]
Now, let's look at the provided response options to see which one correctly represents this:
1. **\(12−−√8\)**: This doesn't correctly express \(8^{\frac{1}{2}}\).
2. **\(\sqrt[2]{8}\)**: This correctly represents \(8^{\frac{1}{2}}\) as the square root of \(8\).
3. **\(8−\sqrt{\frac{1}{2}}\)**: This incorrectly alters the original expression.
4. **\(\sqrt[1]{8}\)**: This is just \(8\) since the first root is the number itself.
5. **\(8−\sqrt{}\)**: This is an incomplete expression and doesn't apply.
From the options provided, the correct way to rewrite \(8^{\frac{1}{2}}\) (interpreted as a square root) is:
**\(\sqrt{8}\)**
or, if your intended notation implies \(\sqrt[2]{8}\), that is also valid, as both refer to the square root of \(8\).
If \(\sqrt[2]{8}\) is represented in your options as **\(\sqrt[2]{8}\)** (though it wasn't formatted in your query), that would also be correct.
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