Question

chapter 6: nth roots and rational exponents > section exercises 6.1 > exercise 26
evaluate the expression using a calculator. round your answer to two decimal places when appropriate.
\sqrt5{1695} \approx \boxed{3.3}

Explanation:

Step1: Recall nth root formula

The nth root of a number \( a \) can be written as \( a^{\frac{1}{n}} \). So, \( \sqrt[5]{1695} = 1695^{\frac{1}{5}} \).

Step2: Calculate using calculator

Using a calculator, we compute \( 1695^{\frac{1}{5}} \). Let's calculate: \( 3.3^5 = 3.3\times3.3\times3.3\times3.3\times3.3 = 3.3^2\times3.3^2\times3.3 = 10.89\times10.89\times3.3 = 118.5921\times3.3 = 391.35393 \), which is not 1695. Wait, maybe a miscalculation. Let's use a calculator properly. \( 1695^{\frac{1}{5}} \approx 5.5 \)? Wait, no, let's do it step by step. Let's find \( x \) such that \( x^5 = 1695 \). Let's try \( 5^5 = 3125 \), which is more than 1695. \( 4^5 = 1024 \), less than 1695. \( 4.5^5 = 4.5\times4.5 = 20.25; 20.25\times4.5 = 91.125; 91.125\times4.5 = 410.0625; 410.0625\times4.5 = 1845.28125 \), which is more than 1695. \( 4.4^5 = 4.4\times4.4 = 19.36; 19.36\times4.4 = 85.184; 85.184\times4.4 = 374.8096; 374.8096\times4.4 = 1649.16224 \), close to 1695. \( 4.42^5 \): Let's compute \( 4.42^2 = 19.5364; 4.42^3 = 19.5364\times4.42 \approx 86.351; 4.42^4 = 86.351\times4.42 \approx 381.671; 4.42^5 = 381.671\times4.42 \approx 1687.0 \). \( 4.43^5 \): \( 4.43^2 = 19.6249; 4.43^3 = 19.6249\times4.43 \approx 86.94 \); \( 4.43^4 = 86.94\times4.43 \approx 385.14 \); \( 4.43^5 = 385.14\times4.43 \approx 1706.17 \). So, using linear approximation between \( x = 4.42 \) ( \( x^5 = 1687 \)) and \( x = 4.43 \) ( \( x^5 = 1706.17 \)). We need \( x^5 = 1695 \). The difference between 1695 and 1687 is 8, and between 1706.17 and 1687 is 19.17. So, \( x \approx 4.42 + \frac{8}{19.17}\times0.01 \approx 4.42 + 0.00417 \approx 4.424 \). Wait, maybe the original problem had a typo? Or maybe I misread the root. Wait, the original problem says \( \sqrt[5]{1695} \approx 3.3 \), but that's incorrect. Wait, maybe it's \( \sqrt[5]{16.95} \)? Let's check \( 3.3^5 = 3.3\times3.3 = 10.89; 10.89\times3.3 = 35.937; 35.937\times3.3 = 118.5921; 118.5921\times3.3 = 391.35393 \), no. Wait, maybe \( \sqrt[5]{1695} \) is actually \( \sqrt[5]{16.95} \)? Let's compute \( \sqrt[5]{16.95} \). \( 2^5 = 32 \), too big. \( 1.7^5 = 1.7\times1.7 = 2.89; 2.89\times1.7 = 4.913; 4.913\times1.7 = 8.3521; 8.3521\times1.7 = 14.19857 \). \( 1.8^5 = 1.8\times1.8 = 3.24; 3.24\times1.8 = 5.832; 5.832\times1.8 = 10.4976; 10.4976\times1.8 = 18.89568 \). So, \( \sqrt[5]{16.95} \) is between 1.7 and 1.8. Wait, the original problem's answer is 3.3, which is wrong. But maybe the problem was \( \sqrt[3]{16.95} \)? \( \sqrt[3]{16.95} \approx 2.56 \), no. Wait, maybe \( \sqrt[4]{1695} \)? \( \sqrt[4]{1695} \approx 6.4 \). Wait, perhaps the user made a mistake in the problem. But assuming we need to correct it, let's use a calculator properly. Let's use a calculator to compute \( 1695^{1/5} \). Using a calculator, \( 1695^{0.2} \approx e^{0.2\ln(1695)} \). \( \ln(1695) \approx 7.44 \), so \( 0.2\times7.44 = 1.488 \), \( e^{1.488} \approx 4.42 \). So the correct value is approximately 4.42 (rounded to two decimal places). But the original answer given is 3.3, which is incorrect. However, if we follow the problem's instruction, maybe there's a miscalculation. Wait, maybe the problem is \( \sqrt[5]{1695} \) and we need to round to two decimal places. Let's use a calculator: \( 1695^{1/5} \approx 4.42 \).

Answer:

The correct approximation of \( \sqrt[5]{1695} \) to two decimal places is approximately \( 4.42 \) (note: the given \( 3.3 \) is incorrect).