How to find the radix (base) of a number given its representation in another radix (base)? What's the method to find the base of any given number?
E.g. find $r$ such that $(121)_r=(144)_8$, where $r$ and $8$ are the bases.
So how do I find the value of $r$?
 A: By definition $$144_8=1\cdot8^2+4\cdot8^1+4\cdot8^0=64+32+4=100\;,$$ and $$121_r=1\cdot r^2+2\cdot r^1+1\cdot r^0=r^2+2r+1\;.$$ To find $r$, just solve the quadratic equation $$r^2+2r+1=100$$ by whatever method you find most convenient; the slickest method is probably to notice that $r^2+2r+1=(r+1)^2$, so $(r+1)^2=100=10^2$.
A: remember the bold letters are the powers

(121)r=(144)8

Now r and 8 is the base so,
$1\cdot r^2+2 \cdot r^1+1\cdot r^0 = 1 \cdot 8^2+4\cdot 8^1+4 \cdot 8^0$

$r^2+2r+1=64+32+4$

$r^2+2r+1=100$

$(r+1)² =100$

$r+1=10$

$r= 10-1$

$r=9$
A: First let's find what 144(8) equals, with "8" as the base
Listing the digits in order, count them off from the right, starting with zero:
digits:     1  4  4
numbering:      2  1  0  
Then the bottom row of numbers become the assigned exponential numbers, and so we have: 
144= 1⋅8ˆ2 + 4⋅8ˆ1 + 4⋅8ˆ0 = 64 + 32 + 4 = 100
As for 121(r), with "r" as the base, we can again list off the digits in order, counting them off from the right starting with zero:
digits:         1  2  1
numbering:      2  1  0
Again, the bottom row of numbers become the assigned exponential numbers, and so we have:
121r= 1⋅rˆ2 + 2⋅rˆ1 + 1⋅rˆ0 = rˆ2 + 2r + 1
We can then find "r" by solving the quadratic equation that we have now created:
rˆ2 + 2r + 1 = 100                            equation
(r + 1)ˆ2 = 100 = 10ˆ2                        factor (r + 1)
± r + 1 = ± 10                                use the square root property
± r = ± 10 - 1                                isolate the "r"
r = 9                                         solve for r
HOPE THIS HELPS!!! JUST CLARIFICATION OF ALL THE STEPS SO IT APPEARS LONGER THAN APPRECIATED :D
