<p><a target="_blank" href="https://www.prepswift.com/quizzes/quiz/prepswift-roots-and-negatives">Roots and Negatives Exercise</a></p><p>In a nutshell, when it comes to <strong><span style="color:#27ae60;">Roots and Negatives</span></strong>, if you're dealing with an even root, you <strong><span style="color:#e74c3c;"><u>cannot</u></span></strong> put a negative number under the radical. Full stop. For example...</p>
<p style="text-align: center;">$\sqrt{-5} =$ <span style="color:#e74c3c;">no good</span> </p>
<p style="text-align: center;">$\sqrt[4]{-11} =$ <span style="color:#e74c3c;">no bueno </span></p>
<p style="text-align: center;">$\sqrt[100!]{-36} =$ <span style="color:#e74c3c;">अच्छा नहीं </span></p>
<p>However, if you're dealing with an odd root, you <strong><span style="color:#27ae60;"><u>can</u> </span></strong>put a negative (or positive) number under the radical. For example...</p>
<p style="text-align: center;">$\sqrt[3]{-5} =$ <span style="color:#27ae60;">no problem</span></p>
<p style="text-align: center;">$\sqrt[5]{-11} =$ <span style="color:#27ae60;">no problema </span></p>
<p style="text-align: center;">$\sqrt[3^{100}]{-36} =$ <span style="color:#27ae60;">कोई बात नहीं </span></p>
<p><span style="color:#8e44ad;"><span style="font-size:20px;">An Important Exception</span></span></p>
<p>If there is an even exponent under the radical, you can do whatever you want. You can put a positive number, a negative number, zero, whatever. This is because you do the exponent operation <strong><u>before</u></strong> the root operation, and the even exponent causes the negative number to become positive. This is why the square root of $x^2$ equals $|x|$. If you don't know what those two lines mean, check out the quant mountain entry on Absolute Value.</p>
<p style="text-align: center;">$$\sqrt{x^2} = |x|$$</p>
<p style="text-align: center;">$\sqrt[6]{(-5)^4}=$ <span style="color:#27ae60;">no problem</span></p>
<p> </p>