Answers:
- Line 2 = Rewrite with a common base
- Line 3 = Power of a Power Property
- Line 4 = Property of Equality
- Max made an error in line 3
- The correct solution is x = 7
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Step-by-step explanation:
Since 25 is equal to 5^2, we use this fact to help get everything as a common base. That common base is 5. That way, we take advantage of the fact that if a^b = a^c, then b = c. In other words, if the two sides are the same, with the same base, then the exponents must be equal as well. This is the property of equality (one form of it).
The power of a power property is used in line 3. This rule is (a^b)^c = a^(b*c). When writing some exponential to another power, we multiply the exponents while keeping the base the same.
Max made a mistake in line 3 when he multiplied the exponents on the left side. The exponents 2 and (x-2) multiply to 2x-4 and not 2x-2. He forgot to distribute fully.
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The equation he needs to solve is 2x-4 = x+3
The steps to solving would be something along the lines of this
2x-4 = x+3
2x-x = 3+4
x = 7
Though there are other paths you could take.
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As a way to check the answer, we plug x = 7 into the original equation and simplify both sides
25^(x-2) = 5^(x+3)
25^(7-2) = 5^(7+3)
25^5 = 5^10
9,765,625 = 9,765,625
We get the same thing on both sides, so this confirms the answer.
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Let's check x = 5 to see how/why it doesn't work
25^(x-2) = 5^(x+3)
25^(5-2) = 5^(5+3)
25^3 = 5^8
15,625 = 390,625
We get two different results on either side. This is a contradiction since the left side says one thing, but the right side says another. This shows that x = 5 is not a solution.