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Radical Misery

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Some time ago I wrote an article about “Exponent Misery”, wherein I discussed some of the confusing things about working with exponents. A next level of confusion is the notation of “radicals” — these symbols \(\sqrt{}\) — and non-integer exponents.

In this article, I’ll go through some of the common hang ups, how to deal with them if you are a student and some pointers on guiding your students if you are in the business of teaching.

The hang ups in a loose chronological order of confusion with attempts at a simple explanation for both student and teacher where necessary:

  1. What the heck is \(\sqrt{}\)?
    • \(\sqrt{}\) is often called the “square root” symbol or more generically a “radical”. If we write \(\sqrt{5}\), we would say it as “square root of 5”. If we write \(\sqrt{x}\), we would say it as “square root of \(x\)”. If we write \(\sqrt{x+1}\) we would say it as “square root of \(x\) plus \(1\)”. Unfortunately, if we write \(\sqrt{x} + 1\), we would also say this as “square root of \(x\), plus \(1\)”. Notice the difference in mathematical notation in the last two examples. When the former is spoken, the \(x + 1\) is said in one breath. When the latter is spoken, there is a pause (indicated by the comma) after the \(x\) before continuing with the “plus \(1\)”.
    • Conceptually and by way of example, the square root of a \(5\) (for example), \(\sqrt{5}\), can be thought of as “what number do I have to square (multiply by itself) to obtain \(5\)?” Remember, that a number doesn’t mean we have to restrict ourselves to the whole numbers. In this case, \(\sqrt{5} \approx 2.23606797749979\); try to compute \(2.23606797749979 \times 2.23606797749979\) (use a calculator) and observe the result — a calculator may give 5.000000000000001 or 4.999999999999999 or 5, depending on its precision.
    • Since we have computing technology at our disposal, build some intuition by evaluating the square root of as many numbers as possible. A classical starting sequence of numbers to try are \(1, 4, 9, 16, 25, 36, 49\).
  2. What is the difference between \(\sqrt{}\) and \(\sqrt[3]{}\) and generally \(\sqrt[n]{}\)?
    • Generally, \(\sqrt[n]{}\) is referred to as the “nth root”. Thus, \(\sqrt[n]{x}\) is said “the nth root of \(x\)”. As an example, \(\sqrt[3]{x}\) is the “third root of \(x\)” often referred to as the “cube root of \(x\)”. A student can understand \(\sqrt{x}\) as the “square root of \(x\)” or equivalently as the “second root of \(x\)”. For all roots beyond three we simply refer to them in their ordinal form. For example, \(\sqrt[5]{x}\) is the “fifth root of \(x\)”.
    • While there is nothing wrong with writing \(\sqrt[2]{x}\), we generally understand that to mean \(\sqrt{x}\). The latter is the preferred standard notation.
    • While there is nothing “wrong” with writing \(\sqrt[\frac{4}{3}]{x}\), this is considered an abuse of notation and is almost never seen in any standard mathematical writing other than, perhaps, in math education writing to demonstrate an abuse (or creative use) of notation. The \(n\) in “nth root”, when written as \(\sqrt[n]{}\) is canonically a positive integer greater than two.
    • If we understand that \(\sqrt{64} = 8\) because \(8 \times 8 = 8^{2} = 64\), then \(\sqrt[3]{64} = 4\) since \(4 \times 4 \times 4 = 4^{3} = 64\). As an exercise to build intuition, and using a calculator, compute square roots, cube roots, and fourth roots of these numbers: \(1, 4, 8, 9, 16, 25, 27, 36, 49, 64, 81, 100, 121, 256, 343\).
  3. I get that \(\sqrt{4} = 2\), but how do I find \(\sqrt{5}\)?
    • There are ways, but typically computing square roots for irrational numbers is no longer done by hand. Instead, we use a calculator. For those curious, here’s an article of mine where I show one way to compute square roots by hand.
    • Typically in an Algebra II class or when square roots are first introduced, students are asked to familiarize themselves with perfect squares as mentioned above. It’s also good practice to try and see what the result is for square roots of these numbers: \(\sqrt{64}, \sqrt{6.4}, \sqrt{.64}, \sqrt{640}, \sqrt{6400}\).
  4. How is \(3^{\frac{1}{2}}\) the same thing as \(\sqrt{3}\)?
    • Ye, old law of exponents come into play. Probably one of the most confusing things for students is translating between radical (\(\sqrt{}\)) to exponent notation (\(\cdot^{n}\)).
    • Recognize that \(\sqrt{3}\) is a different way to write \(3^{\frac{1}{2}}\). Generally, the best way to work with radicals is to get rid of them by converting to exponent notation and working through the mechanics as needed. Then, if there is a requirement, we can convert back to radical notation.
    • Learn the laws of exponents! At the core we have these, though most textbooks give further breakdown.
      • \(x^{m}\cdot x^{n} = x^{m+n}\)
      • \((x^{m})^{n} = x^{m\cdot n }\)
      • \(x^{m}\cdot y^{m} = (x\cdot y)^{m}\)
      • \(x^{-m} = \frac{1}{x^{m}}\)
      • And the above “rules” are enough to figure out what happens with division (and purists would say that I have probably given too many rules as is). Probably the simplest way to to get used to what the rules are is to let \(x = 2, y = 3, m = 5, n = 7\) and substitute. If the right-hand side equals the left-hand side, odds are, you’re in good shape! Otherwise, you’ve probably made a mistake! Here are some examples of the many common mistakes made when applying these rules.
        • \(x^{m} + x^{n} = x^{m+n}\) Not true!
        • \(x^{m} + y^{m} = (x+y)^{m}\) Not true!
        • \(x^{m}\cdot x^{n} = x^{m\cdot n}\) Not true!
        • \(x^{m^{n}} = x^{m\cdot n}\) Not true! (Notice the lack of parentheses and compare against the rule!)
        • \(x^{-m} = -x^{m}\) Not true!
        • \(x^{m}y^{n} = (xy)^{mn}\) Not true!
        • \(\frac{x^{m}}{x^{n}} = x^{\frac{m}{n}}\) Not true!
  5. I don’t know how to write \(7^{\frac{1}{4}}\) as a radical.
    • \(\sqrt[4]{7} = 7^{\frac{1}{4}}\)
  6. I don’t know how to write \(7^{\frac{2}{3}}\) as a radical.
    • \(\sqrt[3]{7^{2}} = (\sqrt[3]{7})^{2}\) since \(7^{\frac{2}{3}} = (7^{2})^{\frac{1}{3}} = (7^\frac{1}{3})^{2}\)
  7. I don’t understand why \(\sqrt{48} = 4\sqrt{3}\).
    • This exercise is typically a root cause for chronic confusion about radicals. As mentioned above, the first thing to do with radicals is to get rid of them. Then, once that’s done, a general way of tackling these problems is to obtain the prime factorization of the number in question (in this case \(48\)) and apply the laws of exponents. Here’s a step-by-step in one line \(\sqrt{48} = 48^{\frac{1}{2}} = (3\cdot 2^{4})^{\frac{1}{2}} = 3^{\frac{1}{2}}\cdot 2^{4\cdot \frac{1}{2}} = 3^{\frac{1}{2}}\cdot 2^{2} = \sqrt{3} \cdot 4 = 4 \cdot \sqrt{3}\). I would recommend that students study this example and explain what rules of exponents allow each step.
  8. Wait, what’s the difference between \(4\sqrt{3}\) and \(\sqrt[4]{3}\)?
    • \(4\sqrt{3}\) is “four times the square root of three” (\(4\cdot 3^{\frac{1}{2}}\)), whereas \(\sqrt[4]{3}\) is “the fourth root of three” (\(3^{\frac{1}{4}}\)).

Ok! You’ve made it this far! Without rewriting a full Algebra chapter on radicals, I hope this was a helpful reference for students and teachers alike.