Question

Express in exponential form
256/81

Answer 1

Answer:

16^2/9^2

more way

4^4/3^4

more and more

2^8/3^4

Answer 2

The multiple exponential representations of the given fractional number are:  $2^{8}/3^{4}$,  $(4/3)^{4}$,   $(1.33)^{4}$.

To convert the given fractional number to an exponential form i.e. in terms of powers, we first do prime factorization of the numerator and the denominator and then form powers by grouping common prime factors.

Prime factorization of numerator :

$256 = 2 * 2*2*2*2*2*2*2$

      $= 2^{8}$

Prime factorization of the denominator :

$81 = 3*3*3*3$

      $= 3^{4}$

Thus, the fraction can be represented in forms of power as :

         $256 / 81 = 2^{8} / 3^{4}$

For complete power exponent on the entire fraction, i.e. numerator and denominator together is :

              $256/81 = 2^{8}/3^{4}$

                          $= (2^{2}) ^{4} / 3^{4}$

                          $= 4^{4}/3^{4}$

                          $= (4/3)^{4}$

                          $= 1.33^{4}$

Thus, 4/3 and 1.33 raised to the power 4 are the fractional and decimal exponential representations of the given fraction.

Hence, $(4/3)^{4}$ and $1.33^{4}$ both are the equivalent exponential representations of the given number.

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