add:%205(0.5k%E2%88%921)+2(0.4%E2%88%92k).%20Answer%20should%20be%20in%20decimal%20form.(1%20point)

Answer 1

To simplify the expression \( 5(0.5k^{-1}) + 2(0.4^{-k}) \), we will clarify each term step-by-step.

1. First Term: \[ 5(0.5k^{-1}) = \frac{5 \cdot 0.5}{k} = \frac{2.5}{k} \]

2. Second Term: \[ 2(0.4^{-k}) = 2 \cdot (0.4^{-k}) = 2 \cdot \frac{1}{0.4^k} \]

So, the original expression becomes: \[ \frac{2.5}{k} + \frac{2}{0.4^k} \]

To write this in decimal form, we recognize that \( 0.4 = \frac{2}{5} \). Thus, \[ 0.4^k = \left(\frac{2}{5}\right)^k \] and therefore, \[ \frac{2}{0.4^k} = \frac{2 \cdot 5^k}{2^k} = \frac{5^k}{2^{k-1}} \]

This leaves us with: \[ \frac{2.5}{k} + \frac{5^k}{2^{k-1}} \]

This expression combines both components. To convert into decimal form, you will need specific values of \( k \) to compute. If you provide a value for \( k \), I can give you the numerical result.