10- Reduza a uma potência de mesma base e simplifique:
A) 32.8^4/4^5
B) 81.9^3/27^2
C) 1000^3.10000/100^2
Soluções para a tarefa
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Vamos lá :
a)
![\frac{32 \times {8}^{4} }{ {4}^{5} } = \frac{ {2}^{5} \times ( { {2}^{3}) }^{4} }{( { {2}^{2}) }^{5} } = \frac{ {2}^{5 } \times {2}^{3 \times 4} }{ {2}^{2 \times 5} } = \frac{ {2}^{5} \times {2}^{12} }{ {2}^{10} } = \\ \\ \frac{ {2}^{5 + 12} }{ {2}^{10} } = \frac{ {2}^{17} }{ {2}^{10} } = {2}^{17 - 10} = {2}^{7} \frac{32 \times {8}^{4} }{ {4}^{5} } = \frac{ {2}^{5} \times ( { {2}^{3}) }^{4} }{( { {2}^{2}) }^{5} } = \frac{ {2}^{5 } \times {2}^{3 \times 4} }{ {2}^{2 \times 5} } = \frac{ {2}^{5} \times {2}^{12} }{ {2}^{10} } = \\ \\ \frac{ {2}^{5 + 12} }{ {2}^{10} } = \frac{ {2}^{17} }{ {2}^{10} } = {2}^{17 - 10} = {2}^{7}](https://tex.z-dn.net/?f=+%5Cfrac%7B32+%5Ctimes++%7B8%7D%5E%7B4%7D+%7D%7B+%7B4%7D%5E%7B5%7D+%7D++%3D++%5Cfrac%7B+%7B2%7D%5E%7B5%7D+%5Ctimes+%28++%7B+%7B2%7D%5E%7B3%7D%29+%7D%5E%7B4%7D++%7D%7B%28+%7B+%7B2%7D%5E%7B2%7D%29+%7D%5E%7B5%7D+%7D++%3D++%5Cfrac%7B+%7B2%7D%5E%7B5+%7D+%5Ctimes++%7B2%7D%5E%7B3+%5Ctimes+4%7D++%7D%7B+%7B2%7D%5E%7B2+%5Ctimes+5%7D+%7D++%3D++%5Cfrac%7B+%7B2%7D%5E%7B5%7D+%5Ctimes++%7B2%7D%5E%7B12%7D+++%7D%7B+%7B2%7D%5E%7B10%7D+%7D++%3D++%5C%5C++%5C%5C++%5Cfrac%7B+%7B2%7D%5E%7B5+%2B+12%7D++%7D%7B+%7B2%7D%5E%7B10%7D+%7D++%3D++%5Cfrac%7B+%7B2%7D%5E%7B17%7D+%7D%7B+%7B2%7D%5E%7B10%7D+%7D+%3D++%7B2%7D%5E%7B17+-+10%7D+++%3D++%7B2%7D%5E%7B7%7D+)
b)
![\frac{81 \times {9}^{3} }{ {27}^{2} } = \frac{ {3}^{4} \times ( { {3}^{2}) }^{3} }{( { {3}^{3}) }^{2} } = \frac{ {3}^{4} \times {3}^{2 \times 3} }{ {3}^{3 \times 2} } = \frac{ {3}^{4} \times {3}^{6} }{ {3}^{6} } = \\ \\ \frac{ {3}^{4 + 6} }{ {3}^{6} } = \frac{ {3}^{10} }{ {3}^{6} } = {3}^{10 - 6} = {3}^{4} \frac{81 \times {9}^{3} }{ {27}^{2} } = \frac{ {3}^{4} \times ( { {3}^{2}) }^{3} }{( { {3}^{3}) }^{2} } = \frac{ {3}^{4} \times {3}^{2 \times 3} }{ {3}^{3 \times 2} } = \frac{ {3}^{4} \times {3}^{6} }{ {3}^{6} } = \\ \\ \frac{ {3}^{4 + 6} }{ {3}^{6} } = \frac{ {3}^{10} }{ {3}^{6} } = {3}^{10 - 6} = {3}^{4}](https://tex.z-dn.net/?f=+%5Cfrac%7B81+%5Ctimes++%7B9%7D%5E%7B3%7D+%7D%7B+%7B27%7D%5E%7B2%7D+%7D++%3D++%5Cfrac%7B++%7B3%7D%5E%7B4%7D+%5Ctimes+%28+%7B+%7B3%7D%5E%7B2%7D%29+%7D%5E%7B3%7D+++%7D%7B%28+%7B+%7B3%7D%5E%7B3%7D%29+%7D%5E%7B2%7D+%7D++%3D++%5Cfrac%7B+%7B3%7D%5E%7B4%7D+%5Ctimes++%7B3%7D%5E%7B2+%5Ctimes+3%7D++%7D%7B+%7B3%7D%5E%7B3+%5Ctimes+2%7D+%7D++%3D++%5Cfrac%7B+%7B3%7D%5E%7B4%7D+%5Ctimes++%7B3%7D%5E%7B6%7D++%7D%7B+%7B3%7D%5E%7B6%7D+%7D++%3D++%5C%5C++%5C%5C++%5Cfrac%7B+%7B3%7D%5E%7B4+%2B+6%7D+%7D%7B+%7B3%7D%5E%7B6%7D+%7D++%3D++%5Cfrac%7B+%7B3%7D%5E%7B10%7D+%7D%7B+%7B3%7D%5E%7B6%7D+%7D++%3D++%7B3%7D%5E%7B10+-+6%7D++%3D++%7B3%7D%5E%7B4%7D+)
c)
![\frac{ {1000}^{3} \times 10000}{ {100}^{2} } = \frac{( { {10}^{3}) }^{3} \times {10}^{4} }{( { {10}^{2}) }^{2} } = \frac{ {10}^{3 \times 3} \times {10}^{4} }{ {10}^{2 \times 2} } = \\ \\ \frac{ {10}^{9} \times {10}^{4} }{ {10}^{4} } = \frac{ {10}^{9 + 4} }{ {10}^{4} } = \frac{ {10}^{13} }{ {10}^{4} } = {10}^{13 - 4} = {10}^{9} \frac{ {1000}^{3} \times 10000}{ {100}^{2} } = \frac{( { {10}^{3}) }^{3} \times {10}^{4} }{( { {10}^{2}) }^{2} } = \frac{ {10}^{3 \times 3} \times {10}^{4} }{ {10}^{2 \times 2} } = \\ \\ \frac{ {10}^{9} \times {10}^{4} }{ {10}^{4} } = \frac{ {10}^{9 + 4} }{ {10}^{4} } = \frac{ {10}^{13} }{ {10}^{4} } = {10}^{13 - 4} = {10}^{9}](https://tex.z-dn.net/?f=+%5Cfrac%7B+%7B1000%7D%5E%7B3%7D++%5Ctimes+10000%7D%7B+%7B100%7D%5E%7B2%7D+%7D++%3D++%5Cfrac%7B%28+%7B+%7B10%7D%5E%7B3%7D%29+%7D%5E%7B3%7D+%5Ctimes++%7B10%7D%5E%7B4%7D++%7D%7B%28+%7B+%7B10%7D%5E%7B2%7D%29+%7D%5E%7B2%7D+%7D++%3D++%5Cfrac%7B+%7B10%7D%5E%7B3+%5Ctimes+3%7D+%5Ctimes++%7B10%7D%5E%7B4%7D++%7D%7B+%7B10%7D%5E%7B2+%5Ctimes+2%7D+%7D++%3D++%5C%5C++%5C%5C++%5Cfrac%7B+%7B10%7D%5E%7B9%7D++%5Ctimes++%7B10%7D%5E%7B4%7D+%7D%7B+%7B10%7D%5E%7B4%7D+%7D++%3D++%5Cfrac%7B+%7B10%7D%5E%7B9+%2B+4%7D+%7D%7B+%7B10%7D%5E%7B4%7D+%7D++%3D++%5Cfrac%7B+%7B10%7D%5E%7B13%7D+%7D%7B+%7B10%7D%5E%7B4%7D+%7D++%3D++%7B10%7D%5E%7B13+-+4%7D++%3D++%7B10%7D%5E%7B9%7D+)
Espero que tenha ajudado.
Bons estudos.
a)
b)
c)
Espero que tenha ajudado.
Bons estudos.
Cintia44:
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