= Solution
For a three-body leptonic decay through the <Fermi interaction>, neglecting daughter <masses> and <radiative corrections>, <lepton universality> gives
$$
\Gamma(\ell^-\to\ell^{\prime-}\bar\nu_{\ell'}\nu_\ell)
=\frac{G_F^2m_\ell^5}{192\pi^3}
$$
in <natural units>. Therefore scaling the <muon decay> rate by $m_\tau^5/m_\mu^5$ predicts only one leptonic channel, not the total <tau lepton> <decay rate>.
The <tau lepton> has two allowed charged-lepton channels, $e^-\bar\nu_e\nu_\tau$ and $\mu^-\bar\nu_\mu\nu_\tau$, as well as hadronic channels from $W^{-*}\to d\bar u,s\bar u$. Each allowed <quark> channel has three <color charge> choices and its <Cabibbo-Kobayashi-Maskawa matrix> factor. Charm production is kinematically inaccessible in a physical hadronic final state at this <mass>. Treating the inclusive hadronic channels by a simple <quark-hadron duality> estimate,
$$
N_{\mathrm{eff}}\simeq2+3(|V_{ud}|^2+|V_{us}|^2)\simeq5.
$$
Thus, using $m_\mu\simeq105.7\,\mathrm{MeV}/c^2$ and $m_\tau\simeq1777\,\mathrm{MeV}/c^2$,
$$
\boxed{\tau_\tau\simeq\frac{2.2\times10^{-6}\,\mathrm s}{5}
\left(\frac{105.7}{1777}\right)^5\simeq3.3\times10^{-13}\,\mathrm s.}
$$
Finite daughter <masses>, hadronic <decay thresholds> and resonances, <Quantum chromodynamics> <radiative corrections>, and electroweak <radiative corrections> refine this estimate. The threefold <color multiplicity in a decay width> and extra decay channels are the main correction to naive fifth-power scaling.
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