Questions: 5^-3 log 5 4+log 5 16+8 log 5 1

Solution Steps

To solve the expression \(5^{-3 \log _{5} 4+\log _{5} 16+8 \log _{5} 1}\), we can use the properties of logarithms and exponents. The key steps are:

1. Simplify each logarithmic term using the properties of logarithms.
2. Combine the terms into a single logarithmic expression.
3. Evaluate the expression using the properties of exponents.

We start by calculating the logarithmic values needed for the expression:

\[ \log_{5} 4 \approx 0.8614 \] \[ \log_{5} 16 \approx 1.7227 \] \[ \log_{5} 1 = 0 \]

Next, we substitute these values into the exponent of the expression:

\[ -3 \log_{5} 4 + \log_{5} 16 + 8 \log_{5} 1 = -3(0.8614) + 1.7227 + 8(0) \]

Calculating this gives:

\[ -3(0.8614) \approx -2.5842 \] \[ -2.5842 + 1.7227 + 0 = -0.8615 \]

Now we evaluate the expression \(5^{-0.8615}\):

\[ 5^{-0.8615} \approx 0.2500 \]

Final Answer