To simplify the expression \(\frac{2×4×8×16}{(log_24)^2(log_48)^3(log_816)^4}\), we will evaluate its components.
First, calculate the individual logarithms in the denominator:
\(log_24 = 2\) as \(2^2 = 4\).
\(log_48\). We know \(8 = 2^3\) and \(log_28 = 3\). Using the change of base formula, \(log_48 = \frac{log_28}{log_24} = \frac{3}{2}\).
\(log_816\). We know \(16 = 2^4\) and \(log_216 = 4\). Using the change of base formula, \(log_816 = \frac{log_216}{log_28} = \frac{4}{3}\).
Substitute these values back into the original expression:
\(\frac{2×4×8×16}{(log_24)^2(log_48)^3(log_816)^4} = \frac{2×4×8×16}{(2)^2(\frac{3}{2})^3(\frac{4}{3})^4}\)
Calculate each part:
Numerator: \(2 × 4 × 8 × 16 = 1024\).
Denominator components: \(2^2 = 4\), \((\frac{3}{2})^3 = \frac{27}{8}\), and \((\frac{4}{3})^4 = \frac{256}{81}\).
Combine the denominator components:
Denominator: \(4 × \frac{27}{8} × \frac{256}{81}\).
Simplify the fraction:
\(\frac{1024}{4 × \frac{27}{8} × \frac{256}{81}} = \frac{1024}{\frac{4 × 27 × 256}{8 × 81}} = \frac{1024}{\frac{27648}{648}}\).
Reduce the fraction in the denominator: \(\frac{27648}{648} = 42.666...\). There appears to be a calculation error in the provided steps. Revisiting the calculation of \(\frac{log_28}{log_24}\) yields \(\frac{3}{2}\). For \(\frac{log_216}{log_28}\) it yields \(\frac{4}{3}\). The calculation of the denominator should be \(4 \times (\frac{3}{2})^3 \times (\frac{4}{3})^4 = 4 \times \frac{27}{8} \times \frac{256}{81}\). Simplifying this gives \( \frac{4 \times 27 \times 256}{8 \times 81} = \frac{1 \times 1 \times 256}{2 \times 3} = \frac{256}{6} = \frac{128}{3}\). Thus, the expression is \(\frac{1024}{\frac{128}{3}} = 1024 \times \frac{3}{128} = 8 \times 3 = 24\).
Therefore, the expression simplifies to:
\(\frac{1024}{24} = \boxed{24}\).
If \[ \log_{p^{1/2}} y \times \log_{y^{1/2}} p = 16, \] then find the value of the given expression.