# Ok I was understanding this but what do powers of base 10 have to do with anything?

• Module 3 Week 2 Day 7 Your Turn Part 2

Ok I was understanding this but what do powers of base 10 have to do with anything?

• @The-Blade-Dancer That's a good question; here Prof. Loh is writing $$11$$ as $$10 + 1.$$ We want

$$11 \times 11 \times 11 \times 11 \times 11 \times 11$$

which is the same as

$$(10 + 1) ( 10 + 1) ( 10 + 1) (10 + 1) (10 + 1) (10 + 1).$$

This will give us terms consisting of integers multiplied to powers of $$10$$ ranging from $$10^0$$ to $$10^6,$$ and we know that multiplying a number by a power of $$10$$ "shifts" it in the place value column. For example, multiplying $$3$$ by $$10^4$$ "shifts" the $$3$$ over by four spots, from the ones column to the ten thousands column.

$$3 \rightarrow 3 \times 10^4 \\ 3 \rightarrow 30,000$$

Now the trick is that the integers multiplied to the power of $$10$$ can be found in a special way: by using Pascal's Triangle!

For the Binomial Theorem that Prof. Loh showed us tells us what the coefficients of the expansion of $$(x+1)^6$$ are:

$$(x + 1)^6 = \binom{6}{0} x^6 1^0 + \binom{6}{1} x^5 1^1 + \binom{6}{2}x^4 1^2 + \binom{6}{3}x^3 1^3 + \binom{6}{4}x^2 1^4 + \binom{6}{5}x^1 1^5 + \binom{6}{6} x^0 1^6$$

These Binomial Coefficients, $$\binom{6}{0}, \binom{6}{1}, \binom{6}{2}, \binom{6}{3}, \binom{6}{4}, \binom{6}{5}, \binom{6}{6},$$ happen to be the values of the 7th row of Pascal's Triangle, and the beautiful thing is that since these Binomial Coefficients are multiplied to successive powers of $$10,$$ we can simply write the result of $$11^6$$ by writing $$\binom{6}{0},$$ then shifting our pen to the left, then writing $$\binom{6}{1},$$ then shifting our pen to the left, then writing $$\binom{6}{2},$$ etc. etc. You could say that these binomial coefficients don't "overlap" each other in our large addition.

\begin{aligned} 1 & \\ 6 \text{ } & \\ 15 \text{ } & \\ 20 \text{ } \text{ } \text{ } & \\ 15 \text{ } \text{ } \text{ }\text{ } \text{ } & \\ 6 \text{ } \text{ } \text{ }\text{ } \text{ } \text{ } \text{ } & \\ 1 \text{ } \text{ } \text{ }\text{ } \text{ } \text{ } \text{ } \text{ } \text{ } & \\ \end{aligned}

If you add these up, you get the answer, $$1771561!$$