Question

\(\frac{2×4×8×16}{(log_24)^2(log_48)^3(log_816)^4}\) equals

Answer 1

We are given: 

\(\log_2 4 = 2,\quad \log_4 8 = \frac{3}{2},\quad \log_8 16 = \frac{4}{3}\)

We have to evaluate:

\[\frac{2 \times 4 \times 8 \times 16}{(\log_2 4)^2 (\log_4 8)^3 (\log_8 16)^4}\]

Now substitute the given values:

\[= \frac{2 \times 4 \times 8 \times 16}{(2)^2 \times \left(\frac{3}{2}\right)^3 \times \left(\frac{4}{3}\right)^4}\]

Calculate the numerator:

\[2 \times 4 \times 8 \times 16 = (2^1)(2^2)(2^3)(2^4) = 2^{1+2+3+4} = 2^{10} = 1024\]

Now compute the denominator step-by-step:

• \((2)^2 = 4\)
• \(\left(\frac{3}{2}\right)^3 = \frac{27}{8}\)
• \(\left(\frac{4}{3}\right)^4 = \frac{256}{81}\)

So, the denominator becomes:

\[4 \times \frac{27}{8} \times \frac{256}{81}\]

Multiply:

\[= \frac{4 \times 27 \times 256}{8 \times 81}\]

Simplify numerator and denominator:

• \(4 \times 27 = 108\)
• \(108 \times 256 = 27648\)
• \(8 \times 81 = 648\)

\[\text{Denominator} = \frac{27648}{648} = 42.666\ldots\]

Now compute final expression: 

\[\frac{1024}{42.666\ldots} = 24\]

Final Answer: \(\boxed{24}\)

Answer 2

We are given: 

$ \log_2 4 = 2,\quad \log_4 8 = \frac{3}{2},\quad \log_8 16 = \frac{4}{3} $

We need to evaluate: $$ \frac{2 \times 4 \times 8 \times 16}{(\log_2 4)^2 \cdot (\log_4 8)^3 \cdot (\log_8 16)^4} $$

Substitute the values into the expression: $$ = \frac{2 \times 4 \times 8 \times 16} {2^2 \cdot \left( \frac{3}{2} \right)^3 \cdot \left( \frac{4}{3} \right)^4} $$

Simplify the denominator step-by-step: 
$2^2 = 4$ 
$\left( \frac{3}{2} \right)^3 = \frac{27}{8}$ 
$\left( \frac{4}{3} \right)^4 = \frac{256}{81}$

Now compute the entire denominator: $$ 4 \cdot \frac{27}{8} \cdot \frac{256}{81} = \frac{4 \cdot 27 \cdot 256}{8 \cdot 81} $$

Simplify numerator: 
$2 \times 4 \times 8 \times 16 = 1024$

So the expression becomes: $$ \frac{1024}{\frac{4 \cdot 27 \cdot 256}{8 \cdot 81}} = \frac{1024 \cdot 8 \cdot 81}{4 \cdot 27 \cdot 256} $$

Cancel common factors and simplify: 
$1024 = 256 \cdot 4$ 
$\Rightarrow$ numerator: $256 \cdot 4 \cdot 8 \cdot 81$ 
denominator: $4 \cdot 27 \cdot 256$ 
Cancel 256 and 4: $$ \frac{8 \cdot 81}{27} = \frac{648}{27} = 24 $$

Final Answer: $24$