44. \( \mathrm{k}= \) ?, \( \mathrm{m}= \) ?, \( \mathrm{n}= \) ? \( \left(10^{2}\right)^{3}=(\sqrt{10})^{k}=\left(10^{m}\right)^{\frac{2}{3}}=\left(\sqrt[5]{10^{3}}\right)^{n} \)

Let’s break down the expression \( \left(10^{2}\right)^{3} \). First, we simplify \( \left(10^{2}\right)^{3} \): \[ \left(10^{2}\right)^{3} = 10^{2 \cdot 3} = 10^{6} \] So, we have \( 10^{6} \). Now, we compare this with \( (\sqrt{10})^{k} \): \[ (\sqrt{10})^{k} = (10^{\frac{1}{2}})^{k} = 10^{\frac{k}{2}} \] Setting the exponents equal gives us: \[ \frac{k}{2} = 6 \Rightarrow k = 12 \] Next, we look at \( \left(10^{m}\right)^{\frac{2}{3}} \): \[ \left(10^{m}\right)^{\frac{2}{3}} = 10^{m \cdot \frac{2}{3}} = 10^{\frac{2m}{3}} \] Setting the exponents equal gives us: \[ \frac{2m}{3} = 6 \Rightarrow 2m = 18 \Rightarrow m = 9 \] Finally, we analyze \( \left(\sqrt[5]{10^{3}}\right)^{n} \): \[ \left(\sqrt[5]{10^{3}}\right)^{n} = (10^{\frac{3}{5}})^{n} = 10^{\frac{3n}{5}} \] Setting the exponents equal gives us: \[ \frac{3n}{5} = 6 \Rightarrow 3n = 30 \Rightarrow n = 10 \] So, the final values are: \[ k = 12, \quad m = 9, \quad n = 10 \]