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Question

Simplify $$\cfrac { { \left( 1024 \right)  }^{ 3 }\times { \left( 81 \right)  }^{ 4 } }{ { \left( 243 \right)  }^{ 2 }\times { \left( 128 \right)  }^{ 4 } } $$


Solution

Observe $$1024={2}^{10}$$, $$81={3}^{4}$$, $$243={3}^{5}$$, and $$128={2}^{7}$$. Hence the expression is
$$\cfrac { { \left( { 2 }^{ 10 } \right)  }^{ 3 }\times { \left( { 3 }^{ 4 } \right)  }^{ 4 } }{ { \left( { 3 }^{ 5 } \right)  }^{ 2 }\times { \left( { 2 }^{ 7 } \right)  }^{ 4 } } =\cfrac { { \left( 2 \right)  }^{ 30 }\times { \left( 3 \right)  }^{ 16 } }{ { \left( 3 \right)  }^{ 10 }\times { \left( 2 \right)  }^{ 28 } } $$
$$={ 2 }^{ 30-28 }\times { 3 }^{ 16-10 }={ 2 }^{ 2 }\times { 3 }^{ 6 }=2916$$

Mathematics

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