Question

sqrt(x) can be written as x^(1/2)
We know (x^(a))^b = x^(ab)

So, (x^2)^(1/2) = x^1 = x

But, we know sqrt(x^2) = |x|
What's this discrepancy about?

Answer 1

\[\sqrt(x) =x ^{1/2} \] and \[(x^a)^b=x ^{ab}\]So, \[(x^{2})^{1/2}=x^1=x\]
But we know that \[\sqrt{x^2}=\left| x \right|\]

Answer 2

whats the question

Answer 3

Why does one method give x, and the other |x| ?

Answer 4

you mean,

\((\sqrt{x} )^2 = x\)

\((\sqrt{x^2} ) = |x|\)

Answer 5

why is it so, is that the question ?

Answer 6

No, not exactly. Because \[\sqrt{x}=x^{1/2}\] so \[\sqrt{x^2}=(x^2)^{1/2}\]

Answer 7

\[(\sqrt{x})^2=(x^{1/2})^2\]

Answer 8

wait

Answer 9

Just the absolute bar is driving me nuts when we write sqrt(x^2) = |x|

Answer 10

whats domain of \((\sqrt{x})^2\) ?

Answer 11

and,

whats domain of \(\sqrt{x^2}\) ?

Answer 12

if u understand why the domains are not same for both above,
you wud see easily why absolute bars are coming for \(\sqrt{x^2}\)

Answer 13

I guess x >=0 and x is real respectively ?

Answer 14

^ right

Answer 15

well I kind of understand it when they use the square root sign. But I get puzzled when we say (x^a)^b= x^(ab), so (x^2)^(1/2) = x which disregards the absolute value.. o.o?

Answer 16

I never saw a prof using absolute values when they wrote square roots as fractions

Answer 17

assume \(x\ge 0\)

then \((x^2)^{1/2}=x\)

and \(((-x)^2)^{1/2}=(x^2)^{1/2}=x\)

so for any \(x\)

\[(x^2)^{1/2}=|x|\]

Answer 18

Also notice,\[\rm \sqrt{(-x)^2} =\left( (-x)^2\right)^{1 \over 2} = (x^2)^{1\over 2} = x \]Now read that backwards. :)