Can someone prove this basic identity?

[Synopoly]

x^(1/n) = the nth root of x

(I'd use mathematical notation but I don't really know how I'm new sorry)

 

[Mentallic]

Try raising both sides to the power of n.

 

[lurflurf]

$$x^{\frac{1}{n}}=\sqrt[n]{x}$$
is usually true by definition as those are two notations for the same function.

 

[Mark44]

$$x^{\frac{1}{n}}=\sqrt[n]{x}$$
is usually true by definition as those are two notations for the same function.

I agree.

 

[rama]

same as what mentallic said,the n(and root n) gets canceled and you are left with the original number
lhs=rhs

 

[jing2178]

$$x^{\frac{1}{n}}=\sqrt[n]{x}$$
is usually true by definition as those are two notations for the same function.

Not so sure that you can just say they are two notations for the same function.

You can define $$\sqrt(x)$$ as that number when multiplied by itself is $$x$$, ie square root is inverse of squaring.

However when dealing with powers you cannot just say well $$x^{\frac{1}{2}}$$ is a new notation for square root, it is the extension of the notation of powers from positive integers to rationals and its interpretation has yet to be determined.
Using the usual rules for powers then

$$x^{\frac{1}{2}}x^{\frac{1}{2}}=x^{{\frac{1}{2}}+{\frac{1}{2}}} =x^1=x$$

it follows that

$$x^{\frac{1}{2}}=\sqrt(x)$$


In the same way you cannot say $$\frac{8}{12}$$ is just another notation for $$\frac{2}{3}$$ without demonstrating their equality

 

[micromass]

Not so sure that you can just say they are two notations for the same function.

You can define $$\sqrt(x)$$ as that number when multiplied by itself is $$x$$, ie square root is inverse of squaring.

However when dealing with powers you cannot just say well $$x^{\frac{1}{2}}$$ is a new notation for square root, it is the extension of the notation of powers from positive integers to rationals and its interpretation has yet to be determined.
Using the usual rules for powers then

$$x^{\frac{1}{2}}x^{\frac{1}{2}}=x^{{\frac{1}{2}}+{\frac{1}{2}}} =x^1=x$$

it follows that

$$x^{\frac{1}{2}}=\sqrt(x)$$


In the same way you cannot say $$\frac{8}{12}$$ is just another notation for $$\frac{2}{3}$$ without demonstrating their equality

Every notation must have a definition. How do you define $x^{1/n}$ and $\sqrt[n]{x}$? Usually, these are defined to coincide.

 

[DrewD]

I personally would think of the rules for exponent to be defined so that $\{a^x|x\in\mathbb{R}\}$ with $a$ being some constant that is greater than zero and not 1, is a group under multiplication. From this, like jing2178 shows, it is easy to prove that $x^{1/n}$ is a number such that, when raised to the $n^{th}$ power, one gets $x$. From there it is clear that the notation $\sqrt[n]{x}$ and $x^{1/n}$ mean the same thing (ignoring discussions of principle roots).

It is equally valid to set the definition that those are the same notation and then show that the exponent rules work out to keep the group structure. To me the other way makes me feel happier inside

 

[micromass]

I personally would think of the rules for exponent to be defined so that $\{a^x|x\in\mathbb{R}\}$ with $a$ being some constant that is greater than zero and not 1, is a group under multiplication. From this, like jing2178 shows, it is easy to prove that $x^{1/n}$ is a number such that, when raised to the $n^{th}$ power, one gets $x$. From there it is clear that the notation $\sqrt[n]{x}$ and $x^{1/n}$ mean the same thing (ignoring discussions of principle roots).

That's actually a neat definition

 

[qspeechc]

$$(x^{1/n})^n = x$$ from which the result follows. E.g. $$\sqrt{4}=2$$ because 2x2=4. $$\sqrt[3]{8}=2$$ because 2x2x2=8.

So if $$(x^{1/n})^n$$ means $$x^{1/n}$$ multiplied by itself n times, and the result gives x, then it must mean $$x^{\frac{1}{n}}=\sqrt[n]{x}$$