Approximate the following number using a calculator.
[tex]$9^{-1.3}$[/tex]
[tex]$9^{-1.3} \approx$[/tex] $\square$ (Round to three decimal places as needed.)
Answer (2)
Calculate 9 − 1.3 .
Round the result to three decimal places.
The approximated value is 0.083 .
Explanation
Understanding the problem We are asked to approximate the value of 9 − 1.3 and round the result to three decimal places.
Calculating the value Using a calculator, we find that 9 − 1.3 ≈ 0.082668...
Rounding the result Rounding this result to three decimal places, we get 0.083 .
Examples
Understanding exponents, especially negative exponents, is crucial in many real-world applications. For instance, in finance, compound interest calculations often involve exponents. If you invest money at a certain interest rate, the future value of your investment can be calculated using the formula A = P ( 1 + r ) n , where A is the future value, P is the principal amount, r is the interest rate, and n is the number of compounding periods. Similarly, negative exponents are used in calculating the present value of a future sum of money. In science, exponents are used to describe exponential decay, such as the decay of radioactive materials. The amount of a radioactive substance remaining after time t is given by N ( t ) = N 0 e − k t , where N 0 is the initial amount, k is the decay constant, and t is the time. Approximating these values helps in making informed decisions in finance and understanding natural phenomena in science.
The approximate value of 9 − 1.3 is 0.090 after calculating 9 1.3 and taking the reciprocal. The rounding to three decimal places gives us the final answer. This process highlights the importance of understanding negative exponents in calculations.
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