Expressing $16b^3a^2(6ab^4)(ab)^3$ in the form $2^m3^na^rb^s$

Question

In Chapter 1.3, Basic Mathematics, Serge Lang, there is the question:

Express the following expression in the form $2^m3^na^rb^s$, where $m, n, r, s$ are positive integers:$$16b^3a^2(6ab^4)(ab)^3$$

The answer I got was $16 \cdot 6 \cdot a^6 \cdot b^{10}$.

Is there something I did wrong?

Answer 1

You're on the right track, just not quite finished:

$16*6=2^4 * 2 * 3=2^5 * 3^1$

Answer 2

Yes, $16\cdot 6$ is not yet in the form $2^m\cdot 3^n$. Otherwise it is correct.

Answer 3

$6$ isn't a power of $3$. You'll want to change $16\cdot 6$ to $32\cdot 3$.

Answer 4

What you've done so far is correct... but we need to do a little more...

$$16 \times 6 = 2^4\times 2 \times 3 = 2^5 \times 3$$

So we have that $$16 \cdot 6 \cdot a^6 \cdot b^{10} = 2^5\cdot 3^1\cdot a^6\cdot b^{10}$$

Answer 5

Great question, believe it or not, these same questions from that same book stumped me at one point.

This might help you out.

Express each of the following expressions in the form $2^m3^na^rb^s$, where $m$, $n$,$ r$ and $ s$ are positive integers.

The goal here is to reduce everything to it's smallest factors. In these questions they all reduce to smallest factors in the form $2^m3^na^rb^s$.