Question

Help with simple small arithmetic.

Answer 1

how does \[\sqrt{x} \times \frac{ 1 }{ x } = \frac{ 1 }{ \sqrt{x} }\] ?

Answer 2

I don't think that I would consider that "simple"

Answer 3

lolz

Answer 4

Well it might be simple for someone a bit older that me.

Answer 5

But that is out of my league lol.

Answer 6

http://www.wolframalpha.com/

Answer 7

a calculator that might help

Answer 8

yea, it shows the result, but I want to know how to simplify it physically.

Answer 9

physically or "by hand"

Answer 10

HI!!

Answer 11

do you know this
\[\sqrt{x}\times \sqrt{x} = x\]

Answer 12

yes

Answer 13

\[\frac{\sqrt{x}}{x}\] is nicely viewed in exponential notation as \[\frac{x^{\frac{1}{2}}}{x}\]

Answer 14

so
\[\frac{ 1 }{ x}= \frac{ 1 }{ \sqrt{x} \times \sqrt{x} }\]

Answer 15

or else start with \[\frac{1}{\sqrt{x}}\times \frac{\sqrt{x}}{\sqrt{x}}=\frac{\sqrt{x}}{x}\]

Answer 16

ok I got it. Thanks all :)

Answer 17

In order to perform this, I have to state to you two useful properties of exponents:
\[1) \frac{ a^n }{ a^m }=a ^{n-m}\]
As a first one, the division between two exponents with the same base "a" is equal to the sustraction of the exponents keeping the same base.
\[(2)\sqrt[n]{a^m}=a ^{\frac{ m }{ n }}\]
The second is, any radical can be expressed as a fractionary exponent, this implying that the radical base is the denominator and the exponent of the inner number "a" is the numerator of the exponent at hand.
So, returning back to the problem in question:
\[\sqrt{x} \times \frac{ 1 }{ x }=\frac{ 1 }{ \sqrt{x} }\]
This is a "proof" excercise, since we want to verify if this equality actually makes sense, so, by the very definition of equality, if I simplify any side as much as I can I should get what's on the other, so, we will limit ourselves in operating the left side: \(\sqrt{x} \times \frac{ 1 }{ x }\):
We operate that fraction, and we that the product of any fraction \(\frac{ a }{ b } \times \frac{ x }{ y }\) is just \(\frac{ ax }{ by }\) so we will apply that:
\[\frac{ \sqrt{x} }{ x }=\frac{ 1 }{ \sqrt{x} }\]
Now, let's apply the second property I stated to you in order to transform that radical into an exponent:
\[\frac{ x ^{\frac{ 1 }{ 2 }} }{ x }=\frac{ 1 }{ x ^{\frac{ 1 }{ 2 }} }\]
And then, let's apply the first property, so we can operate thoe exponents:
\[x ^{\frac{ 1 }{ 2 }-1}=\frac{ 1 }{ x ^{\frac{ 1 }{ 2 }} }\]
Operating the fraction further:
\[x ^{-\frac{ 1 }{ 2 }}=\frac{ 1 }{ x ^{\frac{ 1 }{ 2 }} }\]
And since I forgot, but any negative exponent can be turned negative as long as we put it on the denominator:
\[(3) a ^{-m}=\frac{ 1 }{ a^m }\]
So, applying it:
\[\frac{ 1 }{ x ^{\frac{ 1 }{ 2 }} }=\frac{ 1 }{ x ^{\frac{ 1 }{ 2 }} } \rightarrow \frac{ 1 }{ \sqrt{x} } = \frac{ 1 }{ \sqrt{x} }\]