$\frac{2\times 4\times 8\times 16}{{(}^{{log}_2}\left({log}_{4}8{)}^3\right({log}_{8}16{)}^4}\text{ equals}$

The correct answer is 24.

To find the value of the given expression, we can simplify the numerator and denominator separately.

Step 1: Simplify the Numerator
The numerator is a simple multiplication of numbers:

$\text{Numerator}=2\times 4\times 8\times 16$
Let us compute this product:

$2\times 4=8$
$8\times 8=64$
$64\times 16=1024$
Thus, the numerator is equal to 1024.

Step 2: Simplify the Denominator
The denominator consists of three logarithmic terms raised to different powers:

$\text{Denominator}={\left({\log }_{2}4\right)}^2{\left({\log }_{4}8\right)}^3{\left({\log }_{8}16\right)}^4$
Let us evaluate each term individually using the change of base rule or power rule of logarithms: ${\log }_{a^b}\left(c^d\right)=\frac{d}{b}{\log }_{a}c$.

1) For the first term:

${\log }_{2}4={\log }_2\left(2^2\right)=2$
Raising this to the power of 2:

${\left({\log }_{2}4\right)}^2=2^2=4$

2) For the second term:

${\log }_{4}8={\log }_{2^2}\left(2^3\right)=\frac{3}{2}$
Raising this to the power of 3:

${\left({\log }_{4}8\right)}^3={\left(\frac{3}{2}\right)}^3=\frac{27}{8}$

3) For the third term:

${\log }_{8}16={\log }_{2^3}\left(2^4\right)=\frac{4}{3}$
Raising this to the power of 4:

${\left({\log }_{8}16\right)}^4={\left(\frac{4}{3}\right)}^4=\frac{256}{81}$

Now, multiply these three evaluated terms to find the total denominator:

$\text{Denominator}=4\times \frac{27}{8}\times \frac{256}{81}$
We can simplify this fraction multiplication:

$\text{Denominator}=\left(\frac{4}{8}\right)\times \left(\frac{27}{81}\right)\times 256$
$\text{Denominator}=\frac{1}{2}\times \frac{1}{3}\times 256=\frac{256}{6}=\frac{128}{3}$

Step 3: Calculate the Final Value
Now, substitute the simplified numerator and denominator back into the main expression:

$\text{Value}=\frac{1024}{\left(\frac{128}{3}\right)}$
Multiply the numerator by the reciprocal of the denominator:

$\text{Value}=1024\times \frac{3}{128}$
Since $1024÷128=8$, we have:

$\text{Value}=8\times 3=24$