Simplify the fraction (2^10 + 2^9 + 2^8) / (2^10 + 5 * 2^9).

Step-by-step solution

1. 1

   Do not evaluate the powers. You could compute $1024+512+256=1792$ over $1024+2560=3584$, but the point of the exercise is the technique that still works when the exponents are letters. Factor instead of multiply out.

2. 2

   Factor the smallest power of 2 out of the numerator. The three exponents are $10,9,8$, so the smallest is $2^{8}$:

   $2^{10}+2^{9}+2^{8}=2^{8}\left(2^{2}+2^{1}+1\right)=2^{8}(4+2+1)=7\cdot 2^{8}$

3. 3

   Factor the smallest power of 2 out of the denominator. Here the exponents are $10$ and $9$, so pull out $2^{9}$:

   $2^{10}+5\cdot 2^{9}=2^{9}\left(2+5\right)=7\cdot 2^{9}$

   The coefficient $5$ stays attached to its own term, giving $2+5=7$ inside the bracket.

4. 4

   Cancel the shared factors. Both brackets produced a $7$, and the powers of two subtract:

   $\frac{7\cdot 2^{8}}{7\cdot 2^{9}}=\frac{2^{8}}{2^{9}}=2^{8-9}=2^{-1}=\frac{1}{2}$

5. 5

   Confirm numerically. $\dfrac{1792}{3584}=\dfrac{1}{2}$ exactly, matching the factored result. The appearance of $7$ in both brackets is what makes this fraction collapse so cleanly.