Question
What is the correct simplification of 64 4/3
?(1 point)
Responses
4,096
4,096
256
256
16
16
85.3
?(1 point)
Responses
4,096
4,096
256
256
16
16
85.3
Answers
Answered by
GPT-4o mini
To simplify \( 64 \frac{4}{3} \), you first convert the mixed number to an improper fraction:
1. Convert \( 64 \) to a fraction: \( 64 = \frac{64}{1} \).
2. Combine \( 64 \) and \( \frac{4}{3} \):
\[
64 \frac{4}{3} = \frac{64 \times 3 + 4}{3} = \frac{192 + 4}{3} = \frac{196}{3}
\]
3. If you want a decimal approximation:
\[
\frac{196}{3} \approx 65.33
\]
None of the provided answers (4,096, 256, 16, 85.3) match \( \frac{196}{3} \) or its approximation.
However, if the question refers to \( 64^{\frac{4}{3}} \) instead, then:
1. Calculate \( 64^{\frac{4}{3}} \):
\[
64^{\frac{1}{3}} = 4 \quad (\text{since } 4^3 = 64)
\]
\[
64^{\frac{4}{3}} = (64^{\frac{1}{3}})^4 = 4^4 = 256
\]
The correct simplification of \( 64^{\frac{4}{3}} \) is \( 256 \).
Thus, the correct answer is **256**.
1. Convert \( 64 \) to a fraction: \( 64 = \frac{64}{1} \).
2. Combine \( 64 \) and \( \frac{4}{3} \):
\[
64 \frac{4}{3} = \frac{64 \times 3 + 4}{3} = \frac{192 + 4}{3} = \frac{196}{3}
\]
3. If you want a decimal approximation:
\[
\frac{196}{3} \approx 65.33
\]
None of the provided answers (4,096, 256, 16, 85.3) match \( \frac{196}{3} \) or its approximation.
However, if the question refers to \( 64^{\frac{4}{3}} \) instead, then:
1. Calculate \( 64^{\frac{4}{3}} \):
\[
64^{\frac{1}{3}} = 4 \quad (\text{since } 4^3 = 64)
\]
\[
64^{\frac{4}{3}} = (64^{\frac{1}{3}})^4 = 4^4 = 256
\]
The correct simplification of \( 64^{\frac{4}{3}} \) is \( 256 \).
Thus, the correct answer is **256**.
Answer
What is the correct simplification of 125 4/3
?(1 point)
Responses
25
25
625
625
5
5
125
?(1 point)
Responses
25
25
625
625
5
5
125
Answered by
GPT-4o mini
To simplify \( 125 \frac{4}{3} \), we first convert the mixed number into an improper fraction:
1. Convert \( 125 \) to a fraction: \( 125 = \frac{125}{1} \).
2. Combine \( 125 \) and \( \frac{4}{3} \):
\[
125 \frac{4}{3} = \frac{125 \times 3 + 4}{3} = \frac{375 + 4}{3} = \frac{379}{3}
\]
However, if the question refers to \( 125^{\frac{4}{3}} \) instead, then we'll calculate that:
1. Calculate \( 125^{\frac{1}{3}} \):
\[
125^{\frac{1}{3}} = 5 \quad (\text{since } 5^3 = 125)
\]
2. Then raise that result to the power of 4:
\[
125^{\frac{4}{3}} = (125^{\frac{1}{3}})^4 = 5^4 = 625
\]
So, if the question refers to \( 125^{\frac{4}{3}} \), the correct simplification is **625**. If it refers to the mixed number \( 125 \frac{4}{3} \), then it would be \( \frac{379}{3} \), but that doesn't appear in your options. Therefore, based on the valid options given, the answer would be **625**.
1. Convert \( 125 \) to a fraction: \( 125 = \frac{125}{1} \).
2. Combine \( 125 \) and \( \frac{4}{3} \):
\[
125 \frac{4}{3} = \frac{125 \times 3 + 4}{3} = \frac{375 + 4}{3} = \frac{379}{3}
\]
However, if the question refers to \( 125^{\frac{4}{3}} \) instead, then we'll calculate that:
1. Calculate \( 125^{\frac{1}{3}} \):
\[
125^{\frac{1}{3}} = 5 \quad (\text{since } 5^3 = 125)
\]
2. Then raise that result to the power of 4:
\[
125^{\frac{4}{3}} = (125^{\frac{1}{3}})^4 = 5^4 = 625
\]
So, if the question refers to \( 125^{\frac{4}{3}} \), the correct simplification is **625**. If it refers to the mixed number \( 125 \frac{4}{3} \), then it would be \( \frac{379}{3} \), but that doesn't appear in your options. Therefore, based on the valid options given, the answer would be **625**.
Answer
What is the simplest form of 12513
?(1 point)
Responses
125−−−√
Start Root 125 End Root
5
5
1253−−−−√1
root index 1 Start Root 125 cubed End Root
41.66
?(1 point)
Responses
125−−−√
Start Root 125 End Root
5
5
1253−−−−√1
root index 1 Start Root 125 cubed End Root
41.66
Answered by
GPT-4o mini
To simplify \( 125^{1/3} \) (which is likely what you mean by "12513" in terms of simplifying the expression), we first determine the cube root of 125.
1. **Finding the cube root**:
\[
125^{1/3} = 5 \quad (\text{since } 5^3 = 125)
\]
Now, let's look at the choices you provided:
- \( \sqrt{125} \) (this is the square root, not relevant here)
- 5
- \( \sqrt[3]{125} \) (this is the cube root, which is indeed 5)
- 41.66 (which is not relevant)
The simplest form of \( 125^{1/3} \) is **5**.
Thus, the correct choice from the options you provided is **5**.
1. **Finding the cube root**:
\[
125^{1/3} = 5 \quad (\text{since } 5^3 = 125)
\]
Now, let's look at the choices you provided:
- \( \sqrt{125} \) (this is the square root, not relevant here)
- 5
- \( \sqrt[3]{125} \) (this is the cube root, which is indeed 5)
- 41.66 (which is not relevant)
The simplest form of \( 125^{1/3} \) is **5**.
Thus, the correct choice from the options you provided is **5**.
Answer
Generate an equivalent expression to 136⋅13 −1/2
.(1 point)
Responses
13112
13 superscript Start Fraction 11 over 2 End Fraction baseline
1133
Start Fraction 1 over 13 cubed End Fraction
13−3
13 superscript negative 3 baseline
133
.(1 point)
Responses
13112
13 superscript Start Fraction 11 over 2 End Fraction baseline
1133
Start Fraction 1 over 13 cubed End Fraction
13−3
13 superscript negative 3 baseline
133
Answered by
GPT-4o mini
To generate an equivalent expression to \( 136 \cdot 13^{-\frac{1}{2}} \), we can follow these steps:
1. **Use the property of exponents**:
\[
136 \cdot 13^{-\frac{1}{2}} = \frac{136}{13^{\frac{1}{2}}}
\]
This is because \( a^{-b} = \frac{1}{a^b} \).
2. **Rewrite the expression**:
We know that \( 13^{\frac{1}{2}} = \sqrt{13} \), so:
\[
136 \cdot 13^{-\frac{1}{2}} = \frac{136}{\sqrt{13}}
\]
Now, let’s check which option corresponds to this equivalent expression. Here are the provided choices:
1. \( 13^{\frac{11}{2}} \)
2. \( \frac{1}{13^3} \)
3. \( 13^{-3} \)
4. \( 13^3 \)
None of the options match \( \frac{136}{\sqrt{13}} \) directly; however, we can further analyze them:
For the exponent calculations:
- \( 13^{\frac{11}{2}} \) does not relate.
- \( \frac{1}{13^3} \) does not relate.
- \( 13^{-3} \) does not relate.
- \( 13^{3} \) does not relate.
None of these directly relate to \( 136 \cdot 13^{-\frac{1}{2}} \).
### Conclusion:
The expression \( 136 \cdot 13^{-\frac{1}{2}} \) simplifies to \( \frac{136}{\sqrt{13}} \), which doesn't correspond exactly to the options provided. It appears that none of the options is equivalent to the original expression.
If your question was specifically regarding another format or if there's a calculation I missed, please provide additional details!
1. **Use the property of exponents**:
\[
136 \cdot 13^{-\frac{1}{2}} = \frac{136}{13^{\frac{1}{2}}}
\]
This is because \( a^{-b} = \frac{1}{a^b} \).
2. **Rewrite the expression**:
We know that \( 13^{\frac{1}{2}} = \sqrt{13} \), so:
\[
136 \cdot 13^{-\frac{1}{2}} = \frac{136}{\sqrt{13}}
\]
Now, let’s check which option corresponds to this equivalent expression. Here are the provided choices:
1. \( 13^{\frac{11}{2}} \)
2. \( \frac{1}{13^3} \)
3. \( 13^{-3} \)
4. \( 13^3 \)
None of the options match \( \frac{136}{\sqrt{13}} \) directly; however, we can further analyze them:
For the exponent calculations:
- \( 13^{\frac{11}{2}} \) does not relate.
- \( \frac{1}{13^3} \) does not relate.
- \( 13^{-3} \) does not relate.
- \( 13^{3} \) does not relate.
None of these directly relate to \( 136 \cdot 13^{-\frac{1}{2}} \).
### Conclusion:
The expression \( 136 \cdot 13^{-\frac{1}{2}} \) simplifies to \( \frac{136}{\sqrt{13}} \), which doesn't correspond exactly to the options provided. It appears that none of the options is equivalent to the original expression.
If your question was specifically regarding another format or if there's a calculation I missed, please provide additional details!
Answer
Generate an equivalent expression to 13^6⋅13^−1 devided by 2
.(1 point)
Responses
13112
13 superscript Start Fraction 11 over 2 End Fraction baseline
1133
Start Fraction 1 over 13 cubed End Fraction
13−3
13 superscript negative 3 baseline
133
.(1 point)
Responses
13112
13 superscript Start Fraction 11 over 2 End Fraction baseline
1133
Start Fraction 1 over 13 cubed End Fraction
13−3
13 superscript negative 3 baseline
133
Answered by
GPT-4o mini
To simplify the expression \( \frac{13^6 \cdot 13^{-1}}{2} \), we can follow these steps:
1. **Combine the exponents in the numerator**:
\[
13^6 \cdot 13^{-1} = 13^{6 - 1} = 13^5
\]
2. **Rewrite the expression**:
Now, we can write the expression as:
\[
\frac{13^5}{2}
\]
### Now, let's see which of the provided options is equivalent to this result:
1. \( 13^{\frac{11}{2}} \)
2. \( \frac{1}{13^3} \)
3. \( 13^{-3} \)
4. \( 13^3 \)
None of these options directly match \( \frac{13^5}{2} \).
### Summary
The expression simplifies to \( \frac{13^5}{2} \), and none of the given options are equivalent to this expression. If the question includes additional context or if there’s a specific transformation requested, please provide more details!
1. **Combine the exponents in the numerator**:
\[
13^6 \cdot 13^{-1} = 13^{6 - 1} = 13^5
\]
2. **Rewrite the expression**:
Now, we can write the expression as:
\[
\frac{13^5}{2}
\]
### Now, let's see which of the provided options is equivalent to this result:
1. \( 13^{\frac{11}{2}} \)
2. \( \frac{1}{13^3} \)
3. \( 13^{-3} \)
4. \( 13^3 \)
None of these options directly match \( \frac{13^5}{2} \).
### Summary
The expression simplifies to \( \frac{13^5}{2} \), and none of the given options are equivalent to this expression. If the question includes additional context or if there’s a specific transformation requested, please provide more details!
Answer
Generate an equivalent expression to (5^−6)^−2/3
.(1 point)
Responses
154
Start Fraction 1 over 5 superscript 4 baseline End Fraction
54
5 superscript 4 baseline
5−203
5 superscript Start Fraction negative 20 over 3 End Fraction baseline
5−4
.(1 point)
Responses
154
Start Fraction 1 over 5 superscript 4 baseline End Fraction
54
5 superscript 4 baseline
5−203
5 superscript Start Fraction negative 20 over 3 End Fraction baseline
5−4
Answered by
GPT-4o mini
To simplify the expression \( (5^{-6})^{-2/3} \), we will use the properties of exponents. Here’s how to do it step by step:
1. **Apply the Power of a Power Rule**:
\[
(a^m)^n = a^{m \cdot n}
\]
Thus, we can simplify the expression:
\[
(5^{-6})^{-2/3} = 5^{-6 \cdot (-2/3)} = 5^{4}
\]
Now, we rewrite \( 5^{4} \) in a fraction form if needed:
2. **Convert to Fraction Form**:
\[
5^4 = \frac{1}{5^{-4}} \quad \text{(but in this case, we actually have } 5^4 \text{)}
\]
### Evaluate Given Options:
Now we check the provided options to find an equivalent expression:
1. \( 15^4 \)
2. \( \frac{1}{5^4} \)
3. \( 5^{-20/3} \)
4. \( 5^{-4} \)
None of these options matches \( 5^4 \) directly.
### Summary
The expression \( (5^{-6})^{-2/3} \) simplifies to \( 5^4 \). Since \( 5^4 \) is not listed as an option, the closest option would actually be \( 5^{-4} \) if we were to focus on using a negative exponent (which isn't the intended outcome).
However, the only option that represents \( 5^4 \) in a negative context is neither here. Thus, it seems that **none of the provided options is a direct equivalent expression** to \( 5^4 \).
If there's a specific context or transformation that you had in mind, please let me know!
1. **Apply the Power of a Power Rule**:
\[
(a^m)^n = a^{m \cdot n}
\]
Thus, we can simplify the expression:
\[
(5^{-6})^{-2/3} = 5^{-6 \cdot (-2/3)} = 5^{4}
\]
Now, we rewrite \( 5^{4} \) in a fraction form if needed:
2. **Convert to Fraction Form**:
\[
5^4 = \frac{1}{5^{-4}} \quad \text{(but in this case, we actually have } 5^4 \text{)}
\]
### Evaluate Given Options:
Now we check the provided options to find an equivalent expression:
1. \( 15^4 \)
2. \( \frac{1}{5^4} \)
3. \( 5^{-20/3} \)
4. \( 5^{-4} \)
None of these options matches \( 5^4 \) directly.
### Summary
The expression \( (5^{-6})^{-2/3} \) simplifies to \( 5^4 \). Since \( 5^4 \) is not listed as an option, the closest option would actually be \( 5^{-4} \) if we were to focus on using a negative exponent (which isn't the intended outcome).
However, the only option that represents \( 5^4 \) in a negative context is neither here. Thus, it seems that **none of the provided options is a direct equivalent expression** to \( 5^4 \).
If there's a specific context or transformation that you had in mind, please let me know!