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Chapter 3: Problem 11

Estimate the indicated value without using a calculator. \(\frac{e^{9}}{e^{8.997}}\)

Short Answer

Expert verified
Using the property of exponents, \(\frac{e^{9}}{e^{8.997}} = e^{9-8.997}\). Then, compute the difference, \(9 - 8.997 = 0.003\), yielding \(e^{0.003}\). Since this value is very close to \(e^0\) which equals 1, the estimated value is \(\frac{e^{9}}{e^{8.997}} \approx e^{0.003} \approx 1\).

Step by step solution

01

Apply property of exponents

First, we need to use the property that states \(\frac{a^m}{a^n} = a^{m-n}\), where \(a\) is the base and \(m, n\) are the exponents. In our case, the base is \(e\). So, using this property, we get: \(\frac{e^{9}}{e^{8.997}} = e^{9-8.997}\)
02

Compute the difference

Now, let's compute the difference between the exponents, that is, \(9 - 8.997\). To do this, subtract the numbers: \(9 - 8.997 = 0.003\)
03

Simplify the expression

Replace the subtraction by the result of the previous step: \(e^{9-8.997} = e^{0.003}\)
04

Estimate the value

Since \(e^{0.003}\) is very close to \(e^0\), which equals 1, we can conclude that the expression has a value very close to 1. Thus, the estimated value is: \(\frac{e^{9}}{e^{8.997}} \approx e^{0.003} \approx 1\)

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