Question 508223
Please limit yourself to one question per question.
What you have shown here could be an entire homework assignment.
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{{{7^7 / 7 = 7^(7-1)}}}
{{{7^(7-1)= 7^6}}}
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Visually, you can see that the denominator 7 cancels one of the 7*7*7*7*7*7*7 in the numerator.
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{{{8^2 / 8^5 = 8^(2-5)}}}
{{{8^(2-5)= 8^(-3)}}}
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This brings up another fact about exponents:  
{{{8^(-3) = 1 / 8^3}}}
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{{{2x/2x^3= 1/x^2}}}
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{{{v/3v^2 = 1/3v}}}
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{{{2v^3 / 3uv^3 = 2/3u }}}
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{{{ (100a^8b^(12)) / (10a^5b^(15)) = 10a^3 /b^3}}} 
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{{{ ( (50x^4)(y^3) ) / ( (15x^8)(y) )  =  ( (5*10x^4)(y^3) ) / ( (5*3x^8)(y) )  }}} 
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{{{ ( (5*10x^4)(y^3) ) / ( (5*3x^8)(y) )  =  ( 10y^2) / ( 3x^4 )  }}}