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this is Planck length: $$\ell_p=\sqrt {\frac {G.\hbar}{c^3}} .$$

  1. How much is important the role of this length in the strings theory?

  2. is this planck's length or newton's length! or maybe both of them!?

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up vote 4 down vote accepted

Concerning the second question, the Planck length is the Planck length and not Newton's length (yes, the OP has asked this question). Newton didn't know Planck's constant which was only discovered 2+ centuries later so he could discuss neither Planck's constant nor the Planck length and other natural units which are functions of Planck's constant.

Max Planck realized the importance of Planck's constant $h$ – usually used in the form of the reduced Planck constant i.e. Dirac constant $\hbar=h/2\pi$ – for the black body formula he derived (it appeared previously in high-energy approximations of the black body formula). He was also able to figure out that any quantity (with any units) has a natural unit which may be written as a product of powers of three fundamental universal constants, $\hbar,c,G$. So he derived all the natural units or Planck units such as the Planck length, Planck mass, Planck time, and products of their powers.

What you wrote is the unique product of such powers of $\hbar,c,G$ that has the unit of length: you should be able to verify it is approximately $1.6\times 10^{-35}$ meters. Because it's a length that is calculated purely from constants that are totally natural, and "naturally equal to one", in quantum mechanics ($\hbar$), relativity ($c$), and a theory of gravity ($G$), it's a length that is likely to play a prominent role in any theory that addresses quantum mechanics, relativity, and gravity, i.e. any theory of quantum gravity.

String theory is a theory of quantum gravity (the only known theory of quantum gravity that is free of internal contradictions, in fact), so the Planck length is important in string theory, too. Well, because string theory contains extra dimensions, a more fundamental constant could be a "higher-dimensional Planck length" which is comparable to the usual Planck length you wrote down – but could also be very different if some of the extra dimensions of space are either hugely warped or very large (relatively to the Planck length).

Previous question about the Planck length and my answer about its relevance is here:

How to get Planck length

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