# Equation involving sum of binomial coefficients.

Solve for $$x$$ if $$\sum_{i=0}^{16} {16 \choose i} 5^i = x^8$$

Not sure what to do here. Should I somehow use the binomial theorem to manipulate this to solve for $$x$$, or is there another approach that's better.

• Yes, recognizing that as the Binomial Theorem will make quick work of it. – Matthew Daly Aug 20 at 2:15
• Write out $(1+5)^{16}$ using the Binomial Theorem. – robjohn Aug 20 at 2:16

$$\displaystyle \sum_{i=0}^{16} {16 \choose i} 5^i = x^8$$

or, $$\displaystyle x^8 = \sum_{i=0}^{16} {16 \choose i} 5^i = (5+1)^{16} = 36^8$$

or, $$x^8 - 36^8 = 0$$

or, $$(x-36)(x+36)(x^2+36^2)(x^4+36^4) = 0$$

Now you can find all the eight roots!

More specifically,the eight roots are the following:

$$36 \left(\cos \frac{2 \pi k}{8} + i \sin \frac{2 \pi k}{8} \right)$$ where $$k = 0, \ldots, 7$$

• @Claude Leibovici Fixed... Thanks a lot. – PTDS Aug 20 at 2:54