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605-655 (Medium)|   Algebra|   Exponents|      
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Bunuel
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If ab ≠ 0, does a=b? 08 Feb 2023, 10:12
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sanyasanha
Bunuel please help me to see the mistake in statement 1

x^a=x^b
x^a/x^b=1
x^a-b=1
x^a-b=x^0
a-b=0
a=b
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\(x^{a-b}=1\) does not necessarily mean that a - b = 0.

If x = 1, then a - b can be any number: \(1^{a-b}=1\) is true for all values of a - b.
If x = -1, then a - b can be any EVEN number: \((-1)^{a-b}=1\) is true for all EVEN values of a - b.
Above, one of the posts says that x can be 0 there but 0^0 case is not tested on the GMAT because there is no consensus, as far as I know, about this case. Some say 0^0 = 1, some say 0^0 is not defined.

So, if we knew that x is not 1, -1, or 0, then we could deduce that a - b = 0 but we don't know that for (1), so the first statement is not sufficient.
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sanyasanha
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If ab ≠ 0, does a=b? 08 Feb 2023, 21:14
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Bunuel
sanyasanha
Bunuel please help me to see the mistake in statement 1

x^a=x^b
x^a/x^b=1
x^a-b=1
x^a-b=x^0
a-b=0
a=b

\(x^{a-b}=1\) does not necessarily mean that a - b = 0.

If x = 1, then a - b can be any number: \(1^{a-b}=1\) is true for all values of a - b.
If x = -1, then a - b can be any EVEN number: \((-1)^{a-b}=1\) is true for all EVEN values of a - b.
Above, one of the posts says that x can be 0 there but 0^0 case is not tested on the GMAT because there is no consensus, as far as I know, about this case. Some say 0^0 = 1, some say 0^0 is not defined.

So, if we knew that x is not 1, -1, or 0, then we could deduce that a - b = 0 but we don't know that for (1), so the first statement is not sufficient.
Show more
Understood, thank you!
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