Kenneth Rosen Edition 7 Exercise 6.4 Question 18 (Page No. 421)

Question

Suppose that $b$ is an integer with $b \geq 7.$ Use the binomial theorem and the appropriate row of Pascal’s triangle to find the base-$b$ expansion of $(11)^{4}_{b}$ [that is, the fourth power of the number $(11)_{b}$ in base-$b$ notation].

Answer 1

The  numeral  $11$  in base  $b$  represents the  number  $b+1$.  Therefore the fourth power of this number is  $b^{4}+4b^{3}+6b^{2}+4b+1$  , where the binomial coefficients can be read from Pascal's triangle.  As long as  $b\geq 7$ ,  these coefficients are single digit numbers in base $b$ , so this is the meaning of the numeral $(14641)_{b}$. In short, the numeral formed by concatenating the symbols in the fourth row of Pascal's triangle is the answer.