| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | To solve the equation \(\left(\frac{1}{25}\right)^{x + 2} = 125^{-x}\), we start by expressing both sides with the same base. First, note that: \[ \frac{1}{25} = 25^{-1} = (5^2)^{-1} = 5^{-2} \] and \[ 125 = 5^3 \] Thus, the equation becomes: \[ (5^{-2})^{x + 2} = (5^3)^{-x} \] Using the power of a power property \((a^m)^n = a^{mn}\), we get: \[ 5^{-2(x + 2)} = 5^{3(-x)} \] Simplifying the exponents: \[ 5^{-2x - 4} = 5^{-3x} \] Since the bases are the same, we can set the exponents equal to each other: \[ -2x - 4 = -3x \] Solving for \(x\): \[ -2x - 4 = -3x \] \[ -2x + 3x = 4 \] \[ x = 4 \] Thus, the solution is: \[ \boxed{D} \] |
D |
| deepseekr1 | To solve the equation \(\left(\frac{1}{25}\right)^{x + 2} = 125^{-x}\), express both sides with base \(5\): 1. Rewrite \(\frac{1}{25}\) as \(5^{-2}\) and \(125\) as \(5^3\): \[ \left(5^{-2}\right)^{x + 2} = \left(5^3\right)^{-x} \] 2. Simplify the exponents: \[ 5^{-2(x + 2)} = 5^{-3x} \] 3. Since the bases are equal, equate the exponents: \[ -2(x + 2) = -3x \] 4. Distribute and solve for \(x\): \[ -2x - 4 = -3x \implies x = 4 \] The solution is \(x = 4\), which corresponds to option **D**. Answer: D |
D |
| gpto1 | D | D |
| GPT-4o | To solve the equation \(\left(\frac{1}{25}\right)^{x + 2} = 125^{-x}\), we first express the bases \( \frac{1}{25} \) and \( 125 \) as powers of 5: \(\frac{1}{25} = 25^{-1} = (5^2)^{-1} = 5^{-2}\), and \(125 = 5^3\). Now, substitute these into the original equation: \[ \left(5^{-2}\right)^{x + 2} = \left(5^3\right)^{-x}. \] Next, apply the power rule \((a^m)^n = a^{mn}\) to both sides: \[ 5^{-2(x + 2)} = 5^{3(-x)}. \] Simplify the exponents: \[ 5^{-2x - 4} = 5^{-3x}. \] Since the bases are the same, we can set the exponents equal to each other: \[ -2x - 4 = -3x. \] Solve for \(x\): \[ -2x - 4 = -3x \implies -4 = -x \implies x = 4. \] Thus, the value of \(x\) is 4. Answer: D. |
D |
| llama3.1-70B-Instruct | B The best answer is B. |
D |
| llama3.1-405B-Instruct | A The best answer is A |
D |
| DeepSeek-Coder-V2 | To solve the equation \(\left(\frac{1}{25}\right)^{x + 2} = 125^{-x}\), we start by expressing both sides with the same base. First, note that: \[ \frac{1}{25} = 5^{-2} \quad \text{and} \quad 125 = 5^3 \] Thus, we can rewrite the equation as: \[ (5^{-2})^{x + 2} = (5^3)^{-x} \] Using the power of a power property \((a^m)^n = a^{mn}\), we get: \[ 5^{-2(x + 2)} = 5^{3(-x)} \] Simplifying the exponents: \[ 5^{-2x - 4} = 5^{-3x} \] Since the bases are the same, we can set the exponents equal to each other: \[ -2x - 4 = -3x \] Solving for \(x\): \[ -2x - 4 = -3x \] \[ -2x + 3x = 4 \] \[ x = 4 \] Thus, the solution is: \[ \boxed{D} \] |
D |
| Qwen2-72B-Instruct | B. 2 | D |
| baichuan4 | To solve for $x$, we can use logarithms to get rid of the exponents. We have: $$\left(\frac{1}{25}\right)^{x + 2} = 125^{-x}$$ Rewriting $125$ as $5^3$ and $\frac{1}{25}$ as $5^{-2}$, we get: $$(5^{-2})^{x + 2} = (5^3)^{-x}$$ Using the property of exponents $(a^m)^n = a^{mn}$, we have: $$5^{-2(x + 2)} = 5^{-3x}$$ Since the bases are the same, we can set the exponents equal to each other: $$-2(x + 2) = -3x$$ Solving for $x$, we get: $$-2x - 4 = -3x$$ $$x = 4$$ So, the answer is D. 4. |
D |
| doubao | We can rewrite 125 as 5^3 and 25 as 5^2. So 125^{-x}=(5^3)^{-x}=5^{-3x} and (\frac{1}{25})^{x+2}=(\frac{1}{5^2})^{x+2}=5^{-2(x+2)}=5^{-2x-4}. Then the equation becomes 5^{-2x-4}=5^{-3x}, which means -2x-4=-3x, x=-4+3x, 2x=4, x=2. So the answer is B. |
D |
| minimax | To solve the equation $\left(\frac{1}{25}\right)^{x + 2} = 125^{-x}$, we can express both sides of the equation with the same base. Since $25 = 5^2$ and $125 = 5^3$, we can rewrite the equation as follows: $\left(\frac{1}{5^2}\right)^{x + 2} = \left(5^3\right)^{-x}$ Now, apply the power of a power rule $(a^b)^c = a^{bc}$ to both sides: $\left(5^{-2}\right)^{x + 2} = \left(5^{-3}\right)^{-x}$ This simplifies to: $5^{-2(x + 2)} = 5^{3(-x)}$ Now, since the bases are the same, we can set the exponents equal to each other: $-2(x + 2) = 3(-x)$ Distribute the -2 on the left side: $-2x - 4 = -3x$ Now, solve for x by adding 3x to both sides: $-2x + 3x - 4 = -3x + 3x$ $x - 4 = 0$ Add 4 to both sides: $x = 4$ So the answer is: D. 4 |
D |
| kimi | To solve the equation \(\left(\frac{1}{25}\right)^{x + 2} = 125^{-x}\), we can start by expressing both sides of the equation with the same base. First, recognize that \(\frac{1}{25}\) can be written as \(25^{-1}\) and \(125\) can be written as \(5^3\). So the equation becomes: \[ (25^{-1})^{x + 2} = (5^3)^{-x} \] Now, apply the power of a power rule, which states that \((a^m)^n = a^{mn}\): \[ 25^{-(x + 2)} = 5^{-3x} \] Next, express both sides with the same base by recognizing that \(25 = 5^2\), so \(25^{-1} = (5^2)^{-1} = 5^{-2}\). This gives us: \[ (5^{-2})^{x + 2} = 5^{-3x} \] Now apply the power of a power rule again: \[ 5^{-2(x + 2)} = 5^{-3x} \] Since the bases are the same, we can equate the exponents: \[ -2(x + 2) = -3x \] Expand and solve for \(x\): \[ -2x - 4 = -3x \] Add \(2x\) to both sides: \[ -4 = x \] This result does not match any of the given options (A, B, C, D). It seems there might be a mistake in the original equation or the options provided. However, if we assume the equation is correct and the options are meant to be integers, we can check if there is a simple mistake in the calculation. Let's re-evaluate the steps: \[ 25^{-(x + 2)} = 5^{-3x} \] Since \(25 = 5^2\), we have: \[ (5^2)^{-(x + 2)} = 5^{-3x} \] Which simplifies to: \[ 5^{-2x - 4} = 5^{-3x} \] Equating the exponents: \[ -2x - 4 = -3x \] Solving for \(x\): \[ -2x + 3x = 4 \] \[ x = 4 \] So the correct answer is: Answer: D. 4 |
D |
| wenxinyiyan | $$\left(\frac{1}{25}\right)^{x + 2} = 125^{-x} \implies 25^{-x-2}=5^{-3x} \implies 5^{-2x-4}=5^{-3x} \implies -2x-4=-3x \implies x=4$$ 故答案为 D。 |
D |
| zhipu | To solve the equation \(\left(\frac{1}{25}\right)^{x + 2} = 125^{-x}\), we first express both sides with the same base. We know that: \[ \frac{1}{25} = 25^{-1} \quad \text{and} \quad 125 = 5^3 \] Rewriting the bases: \[ \left(25^{-1}\right)^{x + 2} = (5^3)^{-x} \] Simplify the exponents: \[ 25^{- (x + 2)} = 5^{-3x} \] Since \(25 = 5^2\), we have: \[ (5^2)^{- (x + 2)} = 5^{-3x} \] Simplify the left side: \[ 5^{-2(x + 2)} = 5^{-3x} \] Now, since the bases are the same, we can equate the exponents: \[ -2(x + 2) = -3x \] Solve for \(x\): \[ -2x - 4 = -3x \] Add \(3x\) to both sides: \[ x - 4 = 0 \] Add 4 to both sides: \[ x = 4 \] So, the solution is \(x = 4\). Therefore, the answer is: D. 4 |
D |
| gpto1 | D | D |
| deepseekv3 | To solve the equation: \[ \left(\frac{1}{25}\right)^{x + 2} = 125^{-x} \] **Step 1: Express all terms with the same base.** Both \(25\) and \(125\) are powers of \(5\): \[ 25 = 5^2 \quad \text{and} \quad 125 = 5^3 \] So, rewrite the equation using these bases: \[ \left(\frac{1}{5^2}\right)^{x + 2} = \left(5^3\right)^{-x} \] Simplify the exponents: \[ 5^{-2(x + 2)} = 5^{-3x} \] **Step 2: Since the bases are equal, set the exponents equal to each other.** \[ -2(x + 2) = -3x \] **Step 3: Solve for \(x\).** \[ -2x - 4 = -3x \] Add \(3x\) to both sides: \[ x - 4 = 0 \] Add \(4\) to both sides: \[ x = 4 \] **Final Answer:** \[ \boxed{D} \] |
D |