another question I am stuck on in a practice test. The question is $$\frac{5^{2011} - 5^{2009} +24}{ 5^{2009} +1}$$ Can you cancel out the $5^{2009}$ or not?
Asked
Active
Viewed 39 times
1
-
That would be wishful thinking. – Nov 09 '16 at 07:36
-
Not. Definitely not. – fleablood Nov 09 '16 at 07:41
-
What would you suggest I do? – Nicole Carr Nov 09 '16 at 07:42
-
Canceling $5^{2009}$ out gives $$\frac{25-1+\frac{24}{5^{2009}}}{1+\frac{1}{5^{2009}}}=24.$$ – Michael Hoppe Nov 09 '16 at 07:49
-
Hint: use $a^x - a^y= a^y (a^x-1) $ and $(b^2-1) =(b-1)(b+1) $. Together the answer becomes something very nice. – fleablood Nov 09 '16 at 07:51
-
Hint: $5^2-1=(5+1)(5-1)=6*4=24$. – fleablood Nov 09 '16 at 07:52
4 Answers
1
$\dfrac{5^{2011}-5^{2009}+24}{5^{2009}+1}=$
$\dfrac{5^{2009+2}-5^{2009}+24}{5^{2009}+1}=$
$\dfrac{5^{2009}\cdot5^{2}-5^{2009}+24}{5^{2009}+1}=$
$\dfrac{5^{2009}\cdot25-5^{2009}+24}{5^{2009}+1}=$
$\dfrac{5^{2009}\cdot(25-1)+24}{5^{2009}+1}=$
$\dfrac{5^{2009}\cdot24+24}{5^{2009}+1}=$
$\dfrac{24\cdot(5^{2009}+1)}{5^{2009}+1}=$
$24\cdot\dfrac{5^{2009}+1}{5^{2009}+1}=$
$24$
barak manos
- 43,109
1
You really should have observed that $5^{2011}-5^{2009} = 5^{2009}(5^2-1)$, and take it from there.
With also $24 = 5^2-1$
Pieter21
- 3,475
0
Hint:
$$ \begin{align} \frac{5^{2011} - 5^{2009} +24}{ 5^{2009} +1} & = \frac{5^2(5^{2009}+1) - 5^2 - (5^{2009} + 1) + 1 +24}{ 5^{2009} +1} \\ & = 25 - 1 + \frac{\;\;\cdots\;\;}{5^{2009} +1} \end{align} $$
dxiv
- 76,497