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Chapter 3: Problem 14

Estimate the indicated value without using a calculator. \(\left(\frac{e^{8.0002}}{e^{8}}\right)^{3}\)

Short Answer

Expert verified
The estimated value of the expression \( \left(\frac{e^{8.0002}}{e^{8}}\right)^{3} \) without using a calculator is approximately \(\frac{5003}{5000}\).

Step by step solution

01

Apply properties of exponents

: Recall that: 1. \( a^b / a^c = a^{(b-c)} \), 2. \( (a^b)^c = a^{(b.c)} \) Using these properties, we can simplify the given expression as: \( \left(\frac{e^{8.0002}}{e^{8}}\right)^{3}= \left(e^{(8.0002-8)}\right)^{3} = e^{(3 \times 0.0002)} \)
02

Convert the exponent into a fraction

: Convert the decimal part of the exponent into a fraction to make it easier to estimate: \( e^{(3 \times 0.0002)} = e^{\frac{3}{5000}} \)
03

Estimate the value using the properties of 'e'

: Now, we will estimate the value of 'e' with the given exponent by using the properties of the natural exponent. Since the exponent is very small, we can approximate the value using the first term of the Taylor series expansion of the exponential function: \( e^x \approx 1 + x \) for small values of 'x'. In our case, 'x' is \(\frac{3}{5000}\). Then, we can estimate the expression as: \( e^{\frac{3}{5000}} \approx 1 + \frac{3}{5000} \)
04

Simplify the expression

: Finally, we simplify the expression as: \( 1 + \frac{3}{5000} = \frac{5000+3}{5000} = \frac{5003}{5000} \) So, the estimated value of the expression \( \left(\frac{e^{8.0002}}{e^{8}}\right)^{3} \) without using a calculator is approximately \(\frac{5003}{5000}\).

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Most popular questions from this chapter

Find all numbers \(x\) that satisfy the given equation. \(e^{x}+e^{-x}=6\)

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Estimate the indicated value without using a calculator. \(e^{0.00092}\)
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