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Chapter 3: Problem 13

Estimate the indicated value without using a calculator. \(\left(\frac{e^{7.001}}{e^{7}}\right)^{2}\)

Short Answer

Expert verified
Estimate the indicated value without using a calculator. Solution: The given expression can be simplified as follows: \[\left(\frac{e^{7.001}}{e^{7}}\right)^{2} = \left(e^{(7.001 - 7)}\right)^2 = \left(e^{0.001}\right)^2 = e^{0.001 \cdot 2} = e^{0.002}\] Hence, the estimated value without using a calculator is \(e^{0.002}\).

Step by step solution

01

Simplify the base e terms

First, we want to simplify the base e terms in the numerator and denominator. Recall the law of exponents: \(e^a / e^b = e^{(a-b)}\). Applying this law to our given expression: \(\left(\frac{e^{7.001}}{e^{7}}\right)^{2} = \left(e^{(7.001 - 7)}\right)^2\)
02

Continue simplifying the exponent

Now, we simplify the exponent: \((7.001 - 7) = 0.001\) So the expression becomes: \(\left(e^{0.001}\right)^2\)
03

Apply the power rule

Next, we apply the power rule. Recall the formula \((a^m)^n = a^{m \cdot n}\) for any base a and rational numbers m, n. Using this rule, we get: \(\left(e^{0.001}\right)^2 = e^{0.001 \cdot 2}\)
04

Calculate the final exponent

By multiplying the exponent by 2, we get: \(0.001 \cdot 2 = 0.002\) So, the final expression becomes: \(e^{0.002}\) We have managed to simplify the given expression to \(e^{0.002}\) without using a calculator. This is an estimate of the original expression.

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Most popular questions from this chapter

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