Resolva a expressão exponencial
9^x-2=√27
Soluções para a tarefa
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Olá.
Usaremos as seguintes propriedades:
![\diamondsuit~\Large\boxed{\boxed{\begin{array}{l}
\mathsf{n^{\frac{r}{2}}=\sqrt{n^r}}\\\\
\mathsf{(n^r)^{s}=n^{r\cdot s}}
\end{array}}} \diamondsuit~\Large\boxed{\boxed{\begin{array}{l}
\mathsf{n^{\frac{r}{2}}=\sqrt{n^r}}\\\\
\mathsf{(n^r)^{s}=n^{r\cdot s}}
\end{array}}}](https://tex.z-dn.net/?f=%5Cdiamondsuit%7E%5CLarge%5Cboxed%7B%5Cboxed%7B%5Cbegin%7Barray%7D%7Bl%7D%0A%5Cmathsf%7Bn%5E%7B%5Cfrac%7Br%7D%7B2%7D%7D%3D%5Csqrt%7Bn%5Er%7D%7D%5C%5C%5C%5C%0A%5Cmathsf%7B%28n%5Er%29%5E%7Bs%7D%3Dn%5E%7Br%5Ccdot+s%7D%7D%0A%5Cend%7Barray%7D%7D%7D)
Vamos aos cálculos.
![\Large\begin{array}{l}
\mathsf{9^{x-2}=\sqrt{27}}\\\\
\mathsf{\dfrac{9^{x}}{9^2}=\sqrt{3^3}}\\\\
\mathsf{9^{x}=3^{\frac{3}{2}}\cdot9^2}\\\\
\mathsf{(3^2)^{x}=3^{\frac{3}{2}}\cdot(3^2)^2}\\\\
\mathsf{3^{2x}=3^{\frac{3}{2}}\cdot3^4}\\\\
\mathsf{3^{2x}=3^{\frac{3}{2}+4}}\end{array} \Large\begin{array}{l}
\mathsf{9^{x-2}=\sqrt{27}}\\\\
\mathsf{\dfrac{9^{x}}{9^2}=\sqrt{3^3}}\\\\
\mathsf{9^{x}=3^{\frac{3}{2}}\cdot9^2}\\\\
\mathsf{(3^2)^{x}=3^{\frac{3}{2}}\cdot(3^2)^2}\\\\
\mathsf{3^{2x}=3^{\frac{3}{2}}\cdot3^4}\\\\
\mathsf{3^{2x}=3^{\frac{3}{2}+4}}\end{array}](https://tex.z-dn.net/?f=%5CLarge%5Cbegin%7Barray%7D%7Bl%7D%0A%5Cmathsf%7B9%5E%7Bx-2%7D%3D%5Csqrt%7B27%7D%7D%5C%5C%5C%5C%0A%5Cmathsf%7B%5Cdfrac%7B9%5E%7Bx%7D%7D%7B9%5E2%7D%3D%5Csqrt%7B3%5E3%7D%7D%5C%5C%5C%5C%0A%5Cmathsf%7B9%5E%7Bx%7D%3D3%5E%7B%5Cfrac%7B3%7D%7B2%7D%7D%5Ccdot9%5E2%7D%5C%5C%5C%5C%0A%5Cmathsf%7B%283%5E2%29%5E%7Bx%7D%3D3%5E%7B%5Cfrac%7B3%7D%7B2%7D%7D%5Ccdot%283%5E2%29%5E2%7D%5C%5C%5C%5C%0A%5Cmathsf%7B3%5E%7B2x%7D%3D3%5E%7B%5Cfrac%7B3%7D%7B2%7D%7D%5Ccdot3%5E4%7D%5C%5C%5C%5C%0A%5Cmathsf%7B3%5E%7B2x%7D%3D3%5E%7B%5Cfrac%7B3%7D%7B2%7D%2B4%7D%7D%5Cend%7Barray%7D)
Igualando e nos focando apenas nos expoentes, teremos:
![\mathsf{x=\dfrac{3}{2}+4}\\\\\\
\mathsf{x=\dfrac{3}{2}+\dfrac{2\cdot4}{2}}\\\\\\
\mathsf{x=\dfrac{3}{2}+\dfrac{8}{2}}\\\\\\
\mathsf{x=\dfrac{3+8}{2}}\\\\\\
\boxed{\mathsf{x=\dfrac{11}{2}=5,5}} \mathsf{x=\dfrac{3}{2}+4}\\\\\\
\mathsf{x=\dfrac{3}{2}+\dfrac{2\cdot4}{2}}\\\\\\
\mathsf{x=\dfrac{3}{2}+\dfrac{8}{2}}\\\\\\
\mathsf{x=\dfrac{3+8}{2}}\\\\\\
\boxed{\mathsf{x=\dfrac{11}{2}=5,5}}](https://tex.z-dn.net/?f=%5Cmathsf%7Bx%3D%5Cdfrac%7B3%7D%7B2%7D%2B4%7D%5C%5C%5C%5C%5C%5C%0A%5Cmathsf%7Bx%3D%5Cdfrac%7B3%7D%7B2%7D%2B%5Cdfrac%7B2%5Ccdot4%7D%7B2%7D%7D%5C%5C%5C%5C%5C%5C%0A%5Cmathsf%7Bx%3D%5Cdfrac%7B3%7D%7B2%7D%2B%5Cdfrac%7B8%7D%7B2%7D%7D%5C%5C%5C%5C%5C%5C%0A%5Cmathsf%7Bx%3D%5Cdfrac%7B3%2B8%7D%7B2%7D%7D%5C%5C%5C%5C%5C%5C%0A%5Cboxed%7B%5Cmathsf%7Bx%3D%5Cdfrac%7B11%7D%7B2%7D%3D5%2C5%7D%7D)
Quaisquer dúvidas, deixe nos comentários.
Bons estudos
Usaremos as seguintes propriedades:
Vamos aos cálculos.
Igualando e nos focando apenas nos expoentes, teremos:
Quaisquer dúvidas, deixe nos comentários.
Bons estudos
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