Question

determine the value of base x of (193)x = (623)8

Answer 1

arroximately 26.
Hope it helps!!

Answer 2

The value of the base 'x' is 16.

Given:

             $(193)_\bold{x} = (623)_\bold{8}$

To Find:

            The value of 'x'.±

Solution:

We know that, if two numbers of any number system are equal then they will be equal in the decimal number system.

So the numbers should be converted to the decimal number system in order to find the value of 'x'.

                 $(1*x^2)+(9*x^1)+(3*x^0) = (6*8^2)+(2*8^1)+(3*8^0)$

For converting any number system to the decimal number system, the number of individual places should be multiplied with the positional powers and then added together.

Now,

                                 $~~~x^2+9x+3 = 384+16+3\\~~~~x^2+9x+3=403\\~~~~~~~~~x^2+9x=403-3\\x^2+9x-400=0$

By using the quadratic formula,

                                    $x = \cfrac{-b ~\±\sqrt{b^2-4ac} }{2a}\\\\=\cfrac{-9~\±\sqrt{9^2-4(1)(-400)} }{2*1}\\\\=\cfrac{-9~\±\sqrt{81+1600} }{2}\\\\=\cfrac{-9~\±\sqrt{1681} }{2}\\\\=\cfrac{-9~\±41}{2}$

Two values for 'x' is,

               $=\cfrac{-9+41}{2}~~~~~~~~~~~~~~~~~~~~~~~~~=\cfrac{-9-41}{2}\\=\cfrac{32}{2}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=\cfrac{50}{2}\\\\=16~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~=-25$

As 'x' is a base value that cannot be negative.

Hence the value of the base 'x' is 16.