Let's analyze the problem:

Problem Details:

1. First Expression: $3.1^{1.9}$
2. Second Expression: $\left(\frac{5}{7}\right)^{-0.65}$

Instructions:

• Use a calculator to evaluate these expressions.
• Round the answers to the nearest thousandth (three decimal places).
• Do not round during intermediate computations.

Let’s calculate step-by-step:

1. Evaluate $3.1^{1.9}$:

Using the formula for powers, calculate $3.1^{1.9}$: $3.1^{1.9} \approx 6.652$

Answer: $6.652$ (rounded to the nearest thousandth).

2. Evaluate $\left(\frac{5}{7}\right)^{-0.65}$:

First, compute the base: $\frac{5}{7} = 0.714285714$ Now calculate the power $0.714285714^{-0.65}$: $\approx 1.292$

Answer: $1.292$ (rounded to the nearest thousandth).

Final Results:

1. $3.1^{1.9} = 6.652$
2. $\left(\frac{5}{7}\right)^{-0.65} = 1.292$

Would you like me to explain the calculations in more detail or help with related concepts?

5 Related Questions:

1. What is the general rule for working with fractional exponents?
2. How do negative exponents affect a fraction's value?
3. Can you provide step-by-step guidance on evaluating powers without a calculator?
4. Why is rounding important in these problems?
5. How does the base value influence the result of exponential calculations?

Tip:

Always verify your calculations using a reliable calculator when working with powers and exponents to avoid rounding errors!