SOLUTION: Write each of the following in the indicated bases without multiplying out the various powers: 4 * 5^6 + 11 * 5^4 + 9 in base five

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: Write each of the following in the indicated bases without multiplying out the various powers: 4 * 5^6 + 11 * 5^4 + 9 in base five       Log On


   

Question 348716: Write each of the following in the indicated bases without multiplying out the various powers:
4 * 5^6 + 11 * 5^4 + 9 in base five
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
First, let's look the the powers of 5:
5%5E6 and 5%5E4
In base 5 these numbers would be:
1000000 and 10000
(a 1 followed by a number of zeros that match the exponent)

Now let's look at the other numbers. In base 5 the digits can be 0, 1, 2, 3 or 4. So in base 5, 4 = 4.
But 11 (ten plus 1) is not a valid base 5 number. We have to rewrite it as an expanded, base 5 number:
11 = 2*5 + 1
So:
11 in base 10 = 21 in base 5 (2 fives plus four ones)
9 is also an invalid number in base 5. Rewriting it as an expnaded, base 5 number:
9 = 1*5 + 4
So:
9 in base 10 = 14 in base 5 (1 five plus one one)

So 4%2A5%5E6 in base 5 would be:
4*1000000 = 4000000
and 11%2A5%5E4 in base 5 would be
21 * 10000 = 210000
and 9 in base 5 is
14

Adding all three of these together:
 4000000
  210000
+     14
--------
 4210014

So 4+%2A+5%5E6+%2B+11+%2A+5%5E4+%2B+9 in base 10 is 4210014 in base 5.