Practice test - simplifying expressions

Question

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?

Answer 1

Hint:

$$ \begin{align} \frac{5^{2011} - 5^{2009} +24}{ 5^{2009} +1} & = \frac{5^2(5^{2009}+1) - 5^2 - (5^{2009} + 1) + 1 +24}{ 5^{2009} +1} \\ & = 25 - 1 + \frac{\;\;\cdots\;\;}{5^{2009} +1} \end{align} $$

Answer 2

$5^{2011}-5^{2009}+24=5^{2009}(5^2-1)+24=5^{2009}*24+24=24(5^{2009}+1)$

Answer 3

$\dfrac{5^{2011}-5^{2009}+24}{5^{2009}+1}=$

$\dfrac{5^{2009+2}-5^{2009}+24}{5^{2009}+1}=$

$\dfrac{5^{2009}\cdot5^{2}-5^{2009}+24}{5^{2009}+1}=$

$\dfrac{5^{2009}\cdot25-5^{2009}+24}{5^{2009}+1}=$

$\dfrac{5^{2009}\cdot(25-1)+24}{5^{2009}+1}=$

$\dfrac{5^{2009}\cdot24+24}{5^{2009}+1}=$

$\dfrac{24\cdot(5^{2009}+1)}{5^{2009}+1}=$

$24\cdot\dfrac{5^{2009}+1}{5^{2009}+1}=$

$24$

Answer 4

You really should have observed that $5^{2011}-5^{2009} = 5^{2009}(5^2-1)$, and take it from there.

With also $24 = 5^2-1$