6. Which of the following is equal to $25 \times 15 \times 9 \times 5.4 \times 3.24$ ?
A $3^{9}$
B $3^{10}$
C $3^{11}$
D $3^{14}$
E $3^{17}$
Solution
B
COMMENTARY
The last thing we want to do here is to do the multiplications to work out the value of $25 \times 15 \times 9 \times 5.4 \times 3.24$, and then to factorize the answer.
Instead, we write the decimals as fractions, then factorize the individual numbers, and do some cancellation.

We have $25=5^{2}, 15=3 \times 5,9=3^{2}, 5 \cdot 4=\frac{54}{10}=\frac{27}{5}=\frac{3^{3}}{5}$ and $3.24=\frac{324}{100}=\frac{81}{25}=\frac{3^{4}}{5^{2}}$. Therefore,
$$
25 \times 15 \times 9 \times 5.4 \times 3.24=5^{2} \times(3 \times 5) \times 3^{2} \times \frac{3^{3}}{5} \times \frac{3^{4}}{5^{2}} .
$$

We can now cancel the factors $5^{2}$ and 5 in the numerator and denominator to deduce that
$$
\begin{aligned}
25 \times 15 \times 9 \times 5.4 \times 3.24 & =3 \times 3^{2} \times 3^{3} \times 3^{4} \\
& =3^{10} .
\end{aligned}
$$