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Class 10 Class 12

Simplify : $space space fraction numerator 2 to the power of negative 5 end exponent cross times 10 to the power of negative 5 end exponent cross times 125 over denominator 5 to the power of negative 9 end exponent cross times 6 to the power of negative 5 end exponent end fraction$

Since $125 space equals space 5 space cross times space 5 space cross times space 5 space equals space 5 cubed$

$10 to the power of negative 5 end exponent space equals space left parenthesis 2 cross times 5 right parenthesis to the power of negative 5 end exponent space equals space 2 to the power of negative 5 end exponent space cross times space 5 to the power of negative 5 end exponent 6 to the power of negative 5 end exponent space equals space left parenthesis 2 cross times 3 right parenthesis to the power of negative 5 end exponent space equals space 2 to the power of negative 5 end exponent space cross times space 3 to the power of negative 5 end exponent$

∴ $fraction numerator 2 to the power of negative 5 end exponent cross times 10 to the power of negative 5 end exponent cross times 125 over denominator 5 to the power of negative 9 end exponent space cross times space 6 to the power of negative 5 end exponent end fraction space equals space fraction numerator 2 to the power of negative 5 end exponent cross times 2 to the power of negative 5 end exponent cross times 5 to the power of negative 5 end exponent cross times 5 cubed over denominator 5 to the power of negative 9 end exponent cross times 2 to the power of negative 5 end exponent cross times 3 to the power of negative 5 end exponent end fraction$

 $equals open parentheses 2 to the power of negative 5 end exponent over 2 to the power of negative 5 end exponent close parentheses space cross times space fraction numerator 5 to the power of negative 5 plus left parenthesis negative 5 right parenthesis plus 3 end exponent over denominator 5 to the power of negative 9 end exponent cross times 2 to the power of negative 5 end exponent cross times 3 to the power of negative 5 end exponent end fraction$

 $equals 1 space cross times space fraction numerator 5 to the power of negative 7 end exponent over denominator 5 to the power of negative 9 space end exponent cross times space 3 to the power of negative 5 end exponent end fraction$

 $equals fraction numerator 1 cross times 5 to the power of 9 cross times 3 to the power of 5 over denominator 5 to the power of 7 end fraction$

 $equals 1 cross times 5 to the power of 9 minus 7 end exponent cross times 3 to the power of 5$

 = $1 space cross times space 5 squared space cross times space 3 to the power of 5$

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 = 25 $cross times$ 243 = 6075.

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open parentheses straight x over straight y close parentheses to the power of straight m

is expressed as:

$left parenthesis straight x minus straight y right parenthesis to the power of straight m$
$space straight x to the power of straight m. space straight y to the power of negative straight m end exponent$
$left parenthesis straight x plus straight y right parenthesis to the power of straight m$