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General

General

Easy

Question

Find the value of:
$4 divided by left parenthesis 216 right parenthesis to the power of negative 2 divided by 3 end exponent minus 1 divided by left parenthesis 256 right parenthesis to the power of negative 3 divided by 4 end exponent$

hint Hint:

In Mathematics, lets say, "a" represents a number and " " is the number of times that number is to be multiplied with itself, in order to get a desired result. Then, such result can be written as an, where a is called the base and n is its power/ exponent.
$a to the power of n equals a cross times a cross times horizontal ellipsis cross times a$ n times.

The correct answer is: 80

Step-by-step solution:-
$4 divided by left parenthesis 216 right parenthesis to the power of negative 2 divided by 3 end exponent minus 1 divided by left parenthesis 256 right parenthesis to the power of negative 3 divided by 4 end exponent$
$equals 4 divided by left parenthesis 1 divided by 216 right parenthesis to the power of 2 divided by 3 end exponent minus 1 divided by left parenthesis 1 divided by 256 right parenthesis to the power of 3 divided by 4 end exponent space space space space space space space space space space space space space space space space space space space space text . end text$ $open parentheses b to the power of negative n end exponent equals 1 divided by b to the power of n close parentheses$
$equals 4 divided by open parentheses 1 to the power of 2 divided by 3 end exponent divided by 216 to the power of 2 divided by 3 end exponent close parentheses minus 1 divided by open parentheses 1 to the power of 3 divided by 4 end exponent divided by 256 to the power of 3 divided by 4 end exponent close parentheses horizontal ellipsis midline horizontal ellipsis midline horizontal ellipsis midline horizontal ellipsis midline horizontal ellipsis midline horizontal ellipsis midline horizontal ellipsis midline horizontal ellipsis midline horizontal ellipsis midline horizontal ellipsis midline horizontal ellipsis midline horizontal ellipsis midline horizontal ellipsis midline horizontal ellipsis midline horizontal ellipsis midline horizontal ellipsis midline horizontal ellipsis midline horizontal ellipsis midline horizontal ellipsis midline horizontal ellipsis midline horizontal ellipsis open square brackets a to the power of n divided by b to the power of n divided by b to the power of n equals left parenthesis a divided by b right parenthesis to the power of n close square brackets$
$equals 4 divided by open parentheses 1 divided by 216 to the power of 2 divided by 3 end exponent close parentheses minus 1 divided by open parentheses 1 divided by 256 to the power of 3 divided by 4 end exponent close parentheses$ $open parentheses 1 to the power of n equals 1 close parentheses$
$equals 4 cross times open parentheses 216 to the power of 2 divided by 3 end exponent divided by 1 close parentheses minus 1 cross times open parentheses 256 to the power of 3 divided by 4 end exponent divided by 1 close parentheses$ (Taking reciprocals)
$equals 4 cross times open parentheses 216 to the power of 2 divided by 3 end exponent close parentheses minus 1 to the power of cross times open parentheses 256 to the power of 3 divided by 4 end exponent close parentheses$
$equals 4 to the power of cross times open parentheses 2 cubed cross times 3 cubed close parentheses to the power of 2 divided by 3 end exponent minus 1 to the power of cross times open parentheses 2 to the power of 8 close parentheses to the power of 3 divided by 4 end exponent$ (Simplifying 216 & 256)
$equals 4 cross times open parentheses 2 to the power of 3 cross times 2 divided by 3 end exponent cross times 3 to the power of 3 to the power of cross times 2 divided by 3 end exponent end exponent close parentheses minus 1 to the power of cross times open parentheses 2 to the power of 8 to the power of cross times 3 divided by 4 end exponent end exponent close parentheses$ $open square brackets open parentheses b to the power of n close parentheses to the power of m equals b to the power of n m end exponent close square brackets$
$equals 4 cross times open parentheses 2 squared cross times 3 squared close parentheses minus 1 cross times open parentheses 2 to the power of 6 close parentheses$
$equals 4 to the power of cross times open parentheses 4 to the power of cross times 9 close parentheses minus 1 to the power of cross times left parenthesis 64 right parenthesis$
$equals 4 to the power of cross times left parenthesis 36 right parenthesis minus 64$
=4× (36)-64
=144-64
=80

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