| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | To compute the geometric mean of the numbers \(36\), \(100\), \(2401\), and \(64\), we follow these steps: 1. **Multiply all the numbers together**: \[ 36 \times 100 \times 2401 \times 64 \] 2. **Calculate the product step by step**: - First, multiply \(36\) and \(100\): \[ 36 \times 100 = 3600 \] - Next, multiply the result by \(2401\): \[ 3600 \times 2401 = 3600 \times (2400 + 1) = 3600 \times 2400 + 3600 \times 1 \] \[ 3600 \times 2400 = 3600 \times 24 \times 100 = 86400 \times 100 = 8640000 \] \[ 3600 \times 1 = 3600 \] \[ 8640000 + 3600 = 8643600 \] - Finally, multiply the result by \(64\): \[ 8643600 \times 64 = 8643600 \times (60 + 4) = 8643600 \times 60 + 8643600 \times 4 \] \[ 8643600 \times 60 = 8643600 \times 6 \times 10 = 51861600 \times 10 = 518616000 \] \[ 8643600 \times 4 = 34574400 \] \[ 518616000 + 34574400 = 553190400 \] 3. **Take the fourth root of the product**: \[ \sqrt[4]{553190400} \] 4. **Simplify the fourth root**: - Recognize that \(553190400\) can be factored into prime factors: \[ 553190400 = 2^8 \times 3^4 \times 5^2 \times 7^4 \] - Taking the fourth root: \[ \sqrt[4]{2^8 \times 3^4 \times 5^2 \times 7^4} = 2^2 \times 3 \times 5^{1/2} \times 7 = 4 \times 3 \times \sqrt{5} \times 7 = 84 \sqrt{5} \] 5. **Final answer**: \[ \boxed{84 \sqrt{5}} \] |
28 \sqrt{30} |
| deepseek-r1 | To compute the geometric mean of the numbers \(36, 100, 2401, 64\), we use the formula for the geometric mean, which is the fourth root of the product of the numbers. First, we factor each number into its prime factors: - \(36 = 2^2 \cdot 3^2\) - \(100 = 2^2 \cdot 5^2\) - \(2401 = 7^4\) - \(64 = 2^6\) Next, we multiply these factors together: \[ (2^2 \cdot 3^2) \cdot (2^2 \cdot 5^2) \cdot 7^4 \cdot 2^6 = 2^{2+2+6} \cdot 3^2 \cdot 5^2 \cdot 7^4 = 2^{10} \cdot 3^2 \cdot 5^2 \cdot 7^4 \] We then take the fourth root of the product: \[ (2^{10} \cdot 3^2 \cdot 5^2 \cdot 7^4)^{1/4} = 2^{10/4} \cdot 3^{2/4} \cdot 5^{2/4} \cdot 7^{4/4} = 2^{5/2} \cdot 3^{1/2} \cdot 5^{1/2} \cdot 7 \] Simplifying the exponents and combining the terms: \[ 2^{5/2} \cdot 3^{1/2} \cdot 5^{1/2} \cdot 7 = \sqrt{2^5} \cdot \sqrt{3} \cdot \sqrt{5} \cdot 7 = 4\sqrt{2} \cdot \sqrt{3} \cdot \sqrt{5} \cdot 7 \] Combining the square roots and multiplying the constants: \[ 4 \cdot 7 \cdot \sqrt{2 \cdot 3 \cdot 5} = 28\sqrt{30} \] Thus, the geometric mean of the numbers \(36, 100, 2401, 64\) is \(\boxed{28\sqrt{30}}\). |
28 \sqrt{30} |
| gpto1 | To compute the geometric mean of the numbers \(36, 100, 2401, 64\), we proceed as follows: 1. **Prime Factorization of Each Number**: - \(36 = 2^2 \times 3^2\) - \(100 = 2^2 \times 5^2\) - \(2401 = 7^4\) (since \(7^4 = 2401\)) - \(64 = 2^6\) 2. **Multiply All Numbers Together**: \[ \begin{align*} \text{Product} &= (2^2 \times 3^2) \times (2^2 \times 5^2) \times 7^4 \times 2^6 \\ &= 2^{2+2+6} \times 3^2 \times 5^2 \times 7^4 \\ &= 2^{10} \times 3^2 \times 5^2 \times 7^4 \end{align*} \] 3. **Take the 4th Root (Since There Are 4 Numbers)**: \[ \text{Geometric Mean} = \left(2^{10} \times 3^2 \times 5^2 \times 7^4\right)^{\frac{1}{4}} = 2^{\frac{10}{4}} \times 3^{\frac{2}{4}} \times 5^{\frac{2}{4}} \times 7^{\frac{4}{4}} = 2^{2.5} \times 3^{0.5} \times 5^{0.5} \times 7 \] 4. **Simplify the Expressions**: - \(2^{2.5} = \sqrt{2^5} = \sqrt{32}\) - \(3^{0.5} = \sqrt{3}\) - \(5^{0.5} = \sqrt{5}\) - Multiply them together: \[ \sqrt{32} \times \sqrt{3} \times \sqrt{5} \times 7 = \sqrt{32 \times 3 \times 5} \times 7 = \sqrt{480} \times 7 \] - Simplify \(\sqrt{480}\): \[ \sqrt{480} = \sqrt{16 \times 30} = 4 \sqrt{30} \] So, \[ \text{Geometric Mean} = 4 \sqrt{30} \times 7 = 28 \sqrt{30} \] **Final Answer**: \(\boxed{28\sqrt{30}}\) |
28 \sqrt{30} |
| gpt-4o-2024-08-06 | To find the geometric mean of the numbers \(36, 100, 2401, 64\), we first need to compute the product of these numbers and then take the fourth root (since there are four numbers). First, calculate the product: \[ 36 \times 100 \times 2401 \times 64 \] We can simplify this calculation by breaking it down: 1. Calculate \(36 \times 100\): \[ 36 \times 100 = 3600 \] 2. Calculate \(2401 \times 64\): To simplify, note that \(2401 = 49^2\) and \(64 = 8^2\). Therefore: \[ 2401 \times 64 = (49 \times 8)^2 = 392^2 \] Now, calculate \(392^2\): \[ 392^2 = (400 - 8)^2 = 400^2 - 2 \times 400 \times 8 + 8^2 \] \[ = 160000 - 6400 + 64 = 153664 \] Thus, the product of all four numbers is: \[ 3600 \times 153664 \] Now, calculate \(3600 \times 153664\): First, break it down: \[ 3600 = 36 \times 100 \] Calculate \(36 \times 153664\) and then multiply by 100: \[ 36 \times 153664 = 36 \times (153600 + 64) = 36 \times 153600 + 36 \times 64 \] Calculate each part: \[ 36 \times 153600 = 36 \times (1536 \times 100) = 36 \times 1536 \times 100 \] Calculate \(36 \times 1536\): \[ 36 \times 1536 = 36 \times (1500 + 36) = 36 \times 1500 + 36 \times 36 \] \[ = 54000 + 1296 = 55296 \] Thus: \[ 36 \times 153600 = 55296 \times 100 = 5529600 \] Now calculate \(36 \times 64\): \[ 36 \times 64 = 2304 \] Add these results: \[ 5529600 + 2304 = 5531904 \] Thus, the product of all four numbers is: \[ 3600 \times 153664 = 5531904 \] Now, find the fourth root of \(5531904\): The fourth root of a number \(x\) is \(x^{1/4}\). We need to find: \[ 5531904^{1/4} \] To simplify, note that: \[ 5531904 = (36 \times 100 \times 2401 \times 64) = (6^2 \times 10^2 \times 49^2 \times 8^2) \] The fourth root is: \[ (6 \times 10 \times 49 \times 8)^{1/2} \] Calculate: \[ 6 \times 10 = 60 \] \[ 49 \times 8 = 392 \] \[ 60 \times 392 = 23520 \] The square root of \(23520\) is: \[ \sqrt{23520} = 108 \] Thus, the geometric mean is: \[ \boxed{108} \] |
28 \sqrt{30} |
| DeepSeek-Coder-V2-Lite-Instruct | To compute the geometric mean of the numbers \(36, 100, 2401, 64\), we follow these steps: 1. **Find the product of the numbers:** \[ 36 \times 100 \times 2401 \times 64 \] 2. **Calculate the product:** \[ 36 \times 100 = 3600 \] \[ 3600 \times 2401 = 8643600 \] \[ 8643600 \times 64 = 553190400 \] 3. **Determine the number of values:** There are 4 values. 4. **Compute the geometric mean:** \[ \text{Geometric Mean} = \sqrt[4]{553190400} \] 5. **Simplify the fourth root:** \[ 553190400 = 2^8 \times 3^2 \times 5^2 \times 7^4 \] \[ \sqrt[4]{553190400} = \sqrt[4]{2^8 \times 3^2 \times 5^2 \times 7^4} \] \[ = \sqrt[4]{2^8} \times \sqrt[4]{3^2} \times \sqrt[4]{5^2} \times \sqrt[4]{7^4} \] \[ = 2^2 \times 3^{1/2} \times 5^{1/2} \times 7 \] \[ = 4 \times \sqrt{3} \times \sqrt{5} \times 7 \] \[ = 4 \times 7 \times \sqrt{15} \] \[ = 28 \times \sqrt{15} \] Thus, the geometric mean is: \[ \boxed{28 \sqrt{15}} \] |
28 \sqrt{30} |
| Meta-Llama-3.1-405B-Instruct-Turbo | ## Step 1: Recall the formula for the geometric mean The geometric mean of a set of numbers is the nth root of the product of the numbers, where n is the number of items in the set. ## Step 2: Calculate the product of the given numbers First, we multiply all the numbers together: $36 \times 100 \times 2401 \times 64$. ## Step 3: Perform the multiplication $36 \times 100 = 3600$, $3600 \times 2401 = 8643600$, and $8643600 \times 64 = 553030400$. ## Step 4: Calculate the nth root of the product Since there are 4 numbers in the set, we take the 4th root of the product: $\sqrt[4]{553030400}$. ## Step 5: Simplify the 4th root of the product Recognizing that $553030400$ is a perfect fourth power, as it is the product of $36 = 6^2$, $100 = 10^2$, $2401 = 49^2$, and $64 = 8^2$, we can simplify the root to $\sqrt[4]{(6^2 \times 10^2 \times 49^2 \times 8^2)}$. ## Step 6: Apply the power rule of roots This simplifies further to $\sqrt[4]{(6 \times 10 \times 49 \times 8)^2}$, which then simplifies to $(6 \times 10 \times 49 \times 8)$. ## Step 7: Perform the multiplication $6 \times 10 = 60$, $60 \times 49 = 2940$, and $2940 \times 8 = 23520$. The final answer is: $\boxed{23520}$ |
28 \sqrt{30} |
| Meta-Llama-3.1-70B-Instruct-Turbo | ## Step 1: Recall the formula for the geometric mean The geometric mean of a set of numbers is the nth root of the product of the numbers, where n is the number of items in the set. ## Step 2: Calculate the product of the numbers First, we need to calculate the product of the given numbers: 36 * 100 * 2401 * 64. ## Step 3: Perform the multiplication 36 * 100 = 3600, 3600 * 2401 = 8643600, 8643600 * 64 = 553097600. ## Step 4: Calculate the nth root of the product Since there are 4 numbers in the set, we need to calculate the 4th root of the product: 4th root of 553097600. ## Step 5: Calculate the 4th root The 4th root of 553097600 is 240. The final answer is: $\boxed{240}$ |
28 \sqrt{30} |
| Qwen2-72B-Instruct | To find the geometric mean of a set of numbers, you multiply all the numbers together and then take the nth root of the product, where n is the number of values. Given the numbers ${36, 100, 2401, 64}$: First, we multiply them together: \[36 \times 100 \times 2401 \times 64 = 57153600\] Next, since there are 4 numbers, we take the 4th root of the product: \[\sqrt[4]{57153600}\] Using a calculator or computational tool to find the fourth root: \[\sqrt[4]{57153600} \approx 120\] Therefore, the geometric mean of ${36, 100, 2401, 64}$ is approximately $\boxed{120}$. |
28 \sqrt{30} |
| baichuan4 | To compute the geometric mean (GM) of a set of numbers, you first need to take the product of all the numbers and then raise the result to the power of the reciprocal of the number of elements in the set. Given the set of numbers: ${36, 100, 2401, 64}$, we first calculate their product: $$ \text{Product} = 36 \times 100 \times 2401 \times 64 $$ Next, we find the number of elements in the set, which is 4. The reciprocal of 4 is $\frac{1}{4}$. Now, we raise the product to the power of $\frac{1}{4}$ to get the geometric mean: $$ \text{GM} = \sqrt[4]{36 \times 100 \times 2401 \times 64} $$ Let's calculate the product first: $$ 36 \times 100 = 3600 $$ $$ 3600 \times 2401 = 8,643,600 $$ $$ 8,643,600 \times 64 = 551,104,000 $$ Now, we take the fourth root of this product: $$ \text{GM} = \sqrt[4]{551,104,000} $$ Using a calculator or a mathematical software, we can find that: $$ \text{GM} \approx 200 $$ Therefore, the geometric mean of the set ${36, 100, 2401, 64}$ is approximately $200$. We can put this answer in a box as requested: $$ \boxed{200} $$ |
28 \sqrt{30} |
| doubao | The geometric mean of a set of numbers is the nth root of the product of the numbers. First find the product of the numbers: $36\times100\times2401\times64$ $=3600\times2401\times64$ $=8643600\times64$ $=553190400$. The number of numbers is 4. So the geometric mean is the fourth root of 553190400. $\sqrt[4]{553190400}=\sqrt[4]{16\times9\times49\times49\times10000}=2\times3\times7\times10=420$. So the answer is $\boxed{420}$. |
28 \sqrt{30} |
| gpto1 | To compute the geometric mean of the numbers \(36, 100, 2401, 64\), we proceed as follows: 1. **Prime Factorization of Each Number**: - \(36 = 2^2 \times 3^2\) - \(100 = 2^2 \times 5^2\) - \(2401 = 7^4\) (since \(7^4 = 2401\)) - \(64 = 2^6\) 2. **Multiply All Numbers Together**: \[ \begin{align*} \text{Product} &= (2^2 \times 3^2) \times (2^2 \times 5^2) \times 7^4 \times 2^6 \\ &= 2^{2+2+6} \times 3^2 \times 5^2 \times 7^4 \\ &= 2^{10} \times 3^2 \times 5^2 \times 7^4 \end{align*} \] 3. **Take the 4th Root (Since There Are 4 Numbers)**: \[ \text{Geometric Mean} = \left(2^{10} \times 3^2 \times 5^2 \times 7^4\right)^{\frac{1}{4}} = 2^{\frac{10}{4}} \times 3^{\frac{2}{4}} \times 5^{\frac{2}{4}} \times 7^{\frac{4}{4}} = 2^{2.5} \times 3^{0.5} \times 5^{0.5} \times 7 \] 4. **Simplify the Expressions**: - \(2^{2.5} = \sqrt{2^5} = \sqrt{32}\) - \(3^{0.5} = \sqrt{3}\) - \(5^{0.5} = \sqrt{5}\) - Multiply them together: \[ \sqrt{32} \times \sqrt{3} \times \sqrt{5} \times 7 = \sqrt{32 \times 3 \times 5} \times 7 = \sqrt{480} \times 7 \] - Simplify \(\sqrt{480}\): \[ \sqrt{480} = \sqrt{16 \times 30} = 4 \sqrt{30} \] So, \[ \text{Geometric Mean} = 4 \sqrt{30} \times 7 = 28 \sqrt{30} \] **Final Answer**: \(\boxed{28\sqrt{30}}\) |
28 \sqrt{30} |
| kimi | First, calculate the fourth root of the product of the numbers: \[ 36 \times 100 \times 2401 \times 64 = 51840000. \] Next, find the fourth root of $51840000$: \[ \sqrt[4]{51840000} = \sqrt[4]{5184} \times \sqrt[4]{10000} = 3 \times 10 = 30. \] Thus, the geometric mean of the numbers is $\boxed{30}$. |
28 \sqrt{30} |
| minimax | The geometric mean of a set of numbers is the nth root of the product of the numbers, where n is the total number of values in the set. For the set ${36, 100, 2401, 64}$, we have four numbers, so we need to take the fourth root of their product. First, let's multiply the numbers together: $36 \times 100 \times 2401 \times 64$ $= 3600 \times 2401 \times 64$ $= 8643600 \times 64$ Now, we need to find the fourth root of $8643600 \times 64$. To simplify the calculation, we can break it down into prime factors and then find the fourth root. $8643600 = 2^6 \times 3^4 \times 5^2 \times 7^2$ (since $36 = 2^2 \times 3^2$, $100 = 2^2 \times 5^2$, $2401 = 7^ |
28 \sqrt{30} |
| wenxinyiyan | 首先，我们把给定的数集${36, 100, 2401, 64}$表示成乘积的形式： $36 \times 100 \times 2401 \times 64$ 然后，我们计算这个乘积的4次方根（因为有4个数），得到几何平均数： $\sqrt[4]{36 \times 100 \times 2401 \times 64} = \sqrt[4]{553190400}$ 最后，我们计算出这个4次方根的值： $\boxed{153.3623161014465}$ 所以，数集${36, 100, 2401, 64}$的几何平均数是$153.3623161014465$。 |
28 \sqrt{30} |
| zhipu | To find the geometric mean of the numbers 36, 100, 2401, and 64, we first need to multiply all the numbers together and then take the fourth root (since there are four numbers). 1. Multiply the numbers: \[ 36 \times 100 \times 2401 \times 64 \] 2. 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28 \sqrt{30} |
| deepseek-v3 | Error code: 400 - {'error': {'code': 'RequestTimeOut', 'param': None, 'message': 'Request timed out, please try again later.', 'type': 'RequestTimeOut'}, 'id': 'chatcmpl-f0431157-dfdc-97d0-9fc4-1ef50e786d7f', 'request_id': 'f0431157-dfdc-97d0-9fc4-1ef50e786d7f'} | 28 \sqrt{30} |