find the value of letter $ (x, a, p, \dots) $

(1) $ 4^{2x+3} = 1 $
(2) $ 5^{3-2x} = 5^{-x} $
(3) $ 3^{1-2x} = 243 $
(4) $ 3^{2a} = 3^{-a} $
(5) $ 4^{3x-2} = 1 $
(6) $ 4^{2p} = 4^{-2p-1} $
(7) $ 6^{-2a} = 6^{2-3a} $
(8) $ 2^{2x+2} = 2^{3x} $
(9) $ 6^{3m} \times 6^{-m} = 6^{-2m} $
(10) $ \frac{2^x}{2^x} = 2^{-2x} $

Question

find the value of letter $ (x, a, p, \dots) $

(1) $ 4^{2x+3} = 1 $
(2) $ 5^{3-2x} = 5^{-x} $
(3) $ 3^{1-2x} = 243 $
(4) $ 3^{2a} = 3^{-a} $
(5) $ 4^{3x-2} = 1 $
(6) $ 4^{2p} = 4^{-2p-1} $
(7) $ 6^{-2a} = 6^{2-3a} $
(8) $ 2^{2x+2} = 2^{3x} $
(9) $ 6^{3m} \times 6^{-m} = 6^{-2m} $
(10) $ \frac{2^x}{2^x} = 2^{-2x} $

Accepted Answer

{ "quality": "clear", "question_count": 10, "skills_matched": [ "step_cot", "table_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Exponential Equations", "Exponent Rules" ], "current_concepts": [ "Exponential Equations", "Exponent Rules" ] } ## Solve equations (1) to (3) $$ \begin{aligned} (1)\quad & 4^{2x+3} = 1 \implies 2x + 3 = 0 \implies x = -\frac{3}{2} \ (2)\quad & 5^{3-2x} = 5^{-x} \implies 3 - 2x = -x \implies x = 3 \ (3)\quad & 3^{1-2x} = 243 \implies 3^{1-2x} = 3^5 \implies 1 - 2x = 5 \implies x = -2 \end{aligned} $$ ## Solve equations (4) to (7) $$ \begin{aligned} (4)\quad & 3^{2a} = 3^{-a} \implies 2a = -a \implies a = 0 \ (5)\quad & 4^{3x-2} = 1 \implies 3x - 2 = 0 \implies x = \frac{2}{3} \ (6)\quad & 4^{2p} = 4^{-2p-1} \implies 2p = -2p - 1 \implies p = -\frac{1}{4} \ (7)\quad & 6^{-2a} = 6^{2-3a} \implies -2a = 2 - 3a \implies a = 2 \end{aligned} $$ ## Solve equations (8) to (10) $$ \begin{aligned} (8)\quad & 2^{2x+2} = 2^{3x} \implies 2x + 2 = 3x \implies x = 2 \ (9)\quad & 6^{3m} \times 6^{-m} = 6^{-2m} \implies 6^{2m} = 6^{-2m} \implies 2m = -2m \implies m = 0 \ (10)\quad & \frac{2^x}{2^x} = 2^{-2x} \implies 1 = 2^{-2x} \implies -2x = 0 \implies x = 0 \end{aligned} $$ | No. | Problem | Answer | |---|---|---| | 1 | $4^{2x+3} = 1$ | $x = -\frac{3}{2}$ | | 2 | $5^{3-2x} = 5^{-x}$ | $x = 3$ | | 3 | $3^{1-2x} = 243$ | $x = -2$ | | 4 | $3^{2a} = 3^{-a}$ | $a = 0$ | | 5 | $4^{3x-2} = 1$ | $x = \frac{2}{3}$ | | 6 | $4^{2p} = 4^{-2p-1}$ | $p = -\frac{1}{4}$ | | 7 | $6^{-2a} = 6^{2-3a}$ | $a = 2$ | | 8 | $2^{2x+2} = 2^{3x}$ | $x = 2$ | | 9 | $6^{3m} \times 6^{-m} = 6^{-2m}$ | $m = 0$ | | 10 | $\frac{2^x}{2^x} = 2^{-2x}$ | $x = 0$ | { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Algebra", "Exponential Equations" ] }

No.	Problem	Answer
2	\(5^{3-2x} = 5^{-x}\)	\(x = 3\)
3	\(3^{1-2x} = 243\)	\(x = -2\)
4	\(3^{2a} = 3^{-a}\)	\(a = 0\)
5	\(4^{3x-2} = 1\)	\(x = \frac{2}{3}\)
6	\(4^{2p} = 4^{-2p-1}\)	\(p = -\frac{1}{4}\)
7	\(6^{-2a} = 6^{2-3a}\)	\(a = 2\)
8	\(2^{2x+2} = 2^{3x}\)	\(x = 2\)
9	\(6^{3m} \times 6^{-m} = 6^{-2m}\)	\(m = 0\)
10	\(\frac{2^x}{2^x} = 2^{-2x}\)	\(x = 0\)

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