Inspired by Sandeep Bhardwaj
Statement ( 1 ) : ( − 1 ) − 2 = ( − 1 ) 2 Statement ( 2 ) : ( − 1 ) − 2 3 = ( − 1 ) 2 3
Which of the above statement(s) is/are true?
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5 solutions
How can we say ( − 1 ) 2 3 = i for sure?
( − 1 ) 2 3 = ( − 1 ) 1 + 2 1 = − 1 × − 1 = − i
Or,
( − 1 ) 2 3 = i 3 = − i
Edit: It seems both i & − i are right.
But shouldn't there be a principal value?
Requesting Challenge Master @Calvin Lin
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( − 1 ) 2 3 = ( ( − 1 ) 3 ) 2 1 = ( − 1 ) 2 1 = − 1 = ι
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But what value should we take? How can you be sure that it's i ?
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@Md Omur Faruque – first write -1 =i^2; then simplify. check my solution .i do not know Latex, but i have tried to explain.
@Md Omur Faruque – I also thought the same as S.K, and both ι and − ι seems to be correct.
Calvin explained to me in the report section that as long as the expressions on either side of the equals sign share at least one root then the equation can be considered as correct. In this case both sides can be either i or − i , and hence the equation can be considered as correct, (even though i = − i .) So in this context we don't have to be looking for a principal value; any shared root will do.
statement 2 is not correct. consider L.H.S: (-1)^(-3/2) =( (-1)^1/2)^-3 = i^-3 = i/ (i^4) =i Consider R.H.S: (-1)^(3/2) = ((-1)^1/2)^3 = i^3 = -i now , since L.H.S is not equal to R.H.S, so the statement 2 is not correct , Correct answer is'1 only'.
How does 1/i = -i? Only statement 1 is right
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i 1 = i 2 i = − i [ i 2 = − 1 ]
statement 2 is not correct. consider L.H.S: (-1)^(-3/2) = ((-1)^1/2)^-3 = i^-3= 1/(i^3) = i/(i^4)= i consider R.H.S: (-1)^(3/2) = ((-1)^1/2)^3 =i^3 = (I^2)*i=-i Now since L.H.S is not equal to R.H.S, statement 2 is not correct. answer is 'only1'
On a calculator, (-1)^{3/2}=-i, but (-1)^{-3/2}=i , obviously not equal. Why does the answer make this statement true?
Please, could you check your answer out again?
(-1) ^-3/2 = 1/i
(-1)^3/2 = i
Considering i, the imaginary unit, in other words, i = (-1)^1/2 . Then, (1/i)^-1/2 = -1 , so they are different.
Regards.
J. Claudio.
tnks for this sol
Yes. both the statements are correct. Especially I discuss here the second statement. Rule of complex number is that complex number always occurs with its conjugate. ie in pair, not in single. Therefore LHS=RHS=√-1 = plus or minus i (ie complex conjugate).
Your correction is incorrect.
Yes. both the statements are correct. Especially I discuss here the second statement. Rule of complex number is that complex number always occurs with its conjugate. ie in pair, not in single. Therefore LHS=RHS=√-1 = plus or minus i (ie complex conjugate).
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Your statement, that a complex number always occurs with its conjugate, ONLY APPLIES to solutions of polynomial equations with real coefficients. It DOES NOT apply when evaluating expressions involving i. (By that reasoning, i^3 would be +/- i. It's not.)
Therefore, statement 2, because it is not ALWAYS true, cannot be considered True. EVERYONE who answered 1 only should be marked CORRECT.
Statement 2 is true if you evaluate the power before the root, but it is FALSE if you evaluate the root before the power. (There is no rule requiring a specific order for evaluating rational exponents.)
I request a correction on this problem.
So the site's solution is wrong? I also think both statements are true. How could they not be?
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This does seem a bit ambiguous, I'm not up to date on my mathematical standards, but I can tell you the reason this is coming about. It's because sqrt(a^2) has a solution for ±a, so when you take a=i you will have a valid solution for ±i which is where the confusion is coming from
Statement 1:
● (-1)^(2) = 1
● (-1)^(-2) = 1/ (-1)^2 = 1/1 = 1
Hence Statement 1 is true.
However, looking at Statement 2:
● (-1)^(-3/2) = i^-3 = 1/i^3 (Substituting i^2 = -1) = 1/-i (rationalising denominator) = i/1 = i
● (-1)^(3/2) = i^3 = -i
As -i ≠ i, Statement 2 is incorrect.
Statement 1 is the only correct statement.
statement 2 is not exactly possible... Since it's value is both "i" and "-i" ... Due to that, it can be termed as something that does not exist with proper definition... So, it is only statement 1
Statement 2 is untrue:
( − 1 ) − 3 / 2 = ( ( − 1 ) 3 / 2 ) 1 = ( ( − 1 ) 3 ) 1 = ( − 1 ) 1 = i 1 while ( − 1 ) 3 / 2 = ( − 1 ) 3 = − 1 = i
(-1)^(-3/2) = (i^2)^(-3)/2) =i^(-3)=1/i^3 = 1/(-i)=i: Now (-1)^(3/2 )=(i^2)^(3/2) =i^3=(-i):
That's just another way to get one answer. You could've used − 1 = ( − i ) 2 & that would give the other answer.
Statement 1 : ( − 1 ) − 2 = ( − 1 1 ) 2 = ( − 1 ) 2
Statement 2 : ( − 1 ) 2 − 3 = ( − 1 1 ) 2 3 = ι 1 = ( − 1 ) 2 3 = ι 3 = − ι
∴ Both the statements are correct.