$(-1)^3 = (-1)^{6/2} =  ((-1)^6)^{1/2} = 1^{1/2} = 1$  So it comes $(-1)^3 = 1$  can anybody explain where exactly the mistake in calculation?
43762
I know there is something wrong with this but I don't know where. It's some kind of a math fallacy and it is driving me crazy. Here it is: $$-1= (-1)^3 = (-1)^{6/2} = \sqrt{(-1)^6}= 1?$$
62674
The following question is problem Pinter's Abstract Algebra. And to put things in context: $G$ is a group and $a, b$ are elements of $G$.  I want to show $(ab)^{-1}$ = $b^{-1}a^{-1}$.   I originally thought of proving the fact in the following manner: \begin{align*} (ab)^{-1}(ab) &amp;= e \newline (ab)^{-1}(ab)(b^{-1}) &amp;= (e)(b^{-1}) \newline (ab)^{-1}(a)(bb^{-1}) &amp;= (b^{-1}) \newline (ab)^{-1}(a)(e) &amp;= (b^{-1}) \newline (ab)^{-1}(a) &amp;= (b^{-1}) \newline (ab)^{-1}(a)(a^{-1}) &amp;= (b^{-1})(a^{-1}) \newline (ab)^{-1}(e) &amp;= (b^{-1})(a^{-1}) \newline (ab)^{-1} &amp;= (b^{-1})(a^{-1}) \newline \end{align*}  I know this may seem extremely inefficient to most, and I know there is a shorter way. But would this be considered a legitimate proof?  Thanks in advance!
26390
Is there an error in this question?  Let u = $(1,0,2)$, v = $(2,1,0)$ and w = $(0,2,1)$. Compute  ||u + v - w.||v = v  This is exactly what is in the question (even the dot after w which I don't know what it means).  Thanks
79747
In a ring $R$ with identity 1, show that $$a(-1) = (-1)a = -a \qquad\forall\, a \in R$$ I have started with $a + (-a) = 0$ but cant proceed from here.
110478
If we are given an expression $(-8)^\frac26$, how do we solve it?  If it is $(-8)^\frac13$, we can find the cube root of -8 which is -2. However, if we square it first and find the sixth root, we get +2 (or maybe $\pm2$, which still isn't the same as just -2.  Also realised that when solving logarithmic and exponential questions, I had often made the assumption that if $b^a=b^c$, then $a=c$, either explicitly, or via some formula. I don't know if this is really true now.  So is there some convention by which we solve non-integral powers or is it just undefined to do so or something?  P.s. Talking of non-integral powers, will a quantity like $(-8)^{√3}$, will it be positive or negative. (Basically that's an irrational power now.)
21441
Some one please help me with this question. I am getting -10.5 as the wrong final answer.I found the formula in the following slide.Since the result was coming out in complex number i got its modulus.That is how i ended up with a 10.5 (2 * 5.25).
58567
i have a pretty nasty question. i was glancing through a few olympiad papers and stumbled upon this question:  prove that  $ i = \sqrt {-1}\ $.  i tried the conventional methods namely euler's formula  but could not figure what to do next. how do you actually proceed further? i have checked a few books and a proof is not available.are there any proof available for this
123390
$x^2 + 1 = z = y^3 - 1$  Why $z = 26 $ and only $26$ ? Is there an elementary proof of that ?
107048
If ${n \choose 3} + {n+3-1 \choose 3} = (n)_3$, compute $n$.  Just making sure I am on the right track...  Do I expand each piece out and solve algebraically?
114176
I know that  $\left ( -1 \right )^{2/3}=\left ( \left ( -1 \right )^{2} \right )^{1/3}=1$  But Matlab computes this as $- 0.5 + 0.8660254038i$ a complex number.Why?
40417
The question says to find all the values of $(1+i)^{(1-i)}$  I have trouble figuring out firstly, exactly what values are being looked for. I can toy around with the equation a bit to try to make it look "acceptable" (i.e $ax + byi$ format) but get stuck along the way. So I need help with:  a) figuring out what values are needed. i.e. what does the question $mean$ and some brief background or diagram that explains, in a practical sense, what I'm supposed to be looking for.   b) the algebra that can lead me to a reasonable solution.  --  MY ATTEMPT:  $$ (1+i)^{(1-i)} = (1+i)^{(1-i)}.\frac{(1+i)^{(1+i)}}{(1+i)^{(1+i)}} = \frac{(1+i)^2}{(1+i)^{(1+i)}} = *$$  *is where I get stuck.
3511
$$e^x − 6e^{-x} − 1 = 0$$  No idea how to solve this. If someone could show me the first one or two steps to push me in the right direction that would be great.
51269
$0 = 0 + 0 + 0 + ...$   $0 = (1 - 1) + (1 - 1) + (1 - 1) + ...$   $0 = 1 - 1 + 1 - 1 + 1 - 1 + ...$   $0 = 1 + (-1 + 1) + (-1 + 1) + (-1 + ...$   $0 = 1 + 0 + 0 + 0 + ...$   $0 = 1$   I can't really tell what is obviously wrong with this. It seems to use the same logic as we see in the derivation of things like $\sum_{k=1}^{\infty} k = -\frac{1}{12}$ which appears to be a quirky but accepted fact in mathematics.
79941
With $i^2=-1$, what is $i!$ ?  I'm not sure if it is $i$ or if it is an infinite product $i(i-1)(i-2)(i-3)(i-4)(i-5)\dots$  It would make sense either way.  Is it not defined?
113458
Using the equivalency, I got the answer equal to $1$, however the answer is given as $\frac34$.    also gives $\frac34$.  Could someone tell me that the given answer is wrong or I made a mistake!
114141
Q1 - Why is $\sqrt{-x}*\sqrt{-x}=-x?$  Q2 - I was thinking it would be:  $\sqrt{-x}*\sqrt{-x}=\sqrt{-x*-x}=\sqrt{x^2}$ but apparently not (why not?)  Q3 - What are the formal algebra rules to use? Can I calculate this without using i such as in: $\sqrt{-x}*\sqrt{-x}=i\sqrt{x}*i\sqrt{x}=-\sqrt{x^2}=-x$.
74532
I am confused about something: \begin{eqnarray}     (e^{2 i \pi})^{0.5} = (e^{2 i \pi \cdot 0.5})= e^{i \pi}=-1 \end{eqnarray} but \begin{eqnarray}      e^{2 i \pi}=1~ and~ 1^{0.5}=1  \end{eqnarray} where is my mistake??  thanks