Simplify the expression (2187/16384) × (1024/243)

What you are trying to simplify

You have a product of two fractions: $$\frac{2187}{16384}\times\frac{1024}{243}.$$ A clean way to simplify is to rewrite big numbers as powers of primes so factors cancel quickly.

Rewrite each number as a prime power

• $2187 = 3^7$
• $243 = 3^5$
• $16384 = 2^{14}$
• $1024 = 2^{10}$

So the expression becomes $$\frac{3^7}{2^{14}}\times\frac{2^{10}}{3^5}.$$

Cancel common factors using exponent rules

Group bases together: $$\frac{3^7}{3^5}\times\frac{2^{10}}{2^{14}} = 3^{7-5}\times 2^{10-14} = 3^2\times 2^{-4}.$$ Convert the negative exponent: $$3^2\times 2^{-4} = \frac{3^2}{2^4} = \frac{9}{16}.$$

Quick check

$\frac{9}{16}$ is already in lowest terms because $9$ and $16$ share no common factors.