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Square Roots: Introduction & Simplification (page 1 of 7)

Sections: Square roots, More simplification / Multiplication, Adding (and subtracting) square roots, Conjugates / Dividing by square roots, Rationalizing denominators, Higher-Index Roots, A special case of rationalizing / Radicals & exponents / Radicals & domains

"Roots" (or "radicals") are the "opposite" operation of applying exponents; you can "undo" a power with a radical, and a radical can "undo" a power. For instance, if you square 2, you get 4, and if you "take the square root of 4", you get 2; if you square 3, you get 9, and if you "take the square root of 9", you get 3:   Copyright © Elizabeth Stapel 1999-2011 All Rights Reserved

    2^2 = 4, so sqrt(4) = 2; 3^2 = 9, so sqrt(9) = 3

The "radical symbol" symbol is called the "radical"symbol. (Technically, just the "check mark" part of the symbol is the radical; the line across the top is called the "vinculum".) The expression " sqrt[9] " is read as "root nine", "radical nine", or "the square root of nine".

You can raise numbers to powers other than just 2; you can cube things, raise them to the fourth power, raise them to the 100th power, and so forth. In the same way, you can take the cube root of a number, the fourth root, the 100th root, and so forth. To indicate some root other than a square root, you use the same radical symbol, but you insert a number into the radical, tucking it into the "check mark" part. For instance:

    4^3 = 64, so the cube root of 64 equals 4

The "3" in the above is the "index" of the radical; the "64" is "the argument of the radical", also called "the radicand". Since most radicals you see are square roots, the index is not included on square roots. While " square root symbol with '2' as index " would be technically correct, I've never seen it used.




    a square (second) root is written as radical symbol

    a cube (third) root is written as cbrt()

    a fourth root is written as  fourth-root()

    a fifth root is written as:  fifth-root()

You can take any counting number, square it, and end up with a nice neat number. But the process doesn't always work going backwards. For instance, consider sqrt(3), the square root of three. There is no nice neat number that squares to 3, so sqrt(3) cannot be simplified as a nice whole number. You can deal with sqrt(3) in either of two ways: If you are doing a word problem and are trying to find, say, the rate of speed, then you would grab your calculator and find the decimal approximation of sqrt(3):

    sqrt(3) = 1.732050808 (approx)

Then you'd round the above value to an appropriate number of decimal places and use a real-world unit or label, like "1.7 ft/sec". On the other hand, you may be solving a plain old math exercise, something with no "practical" application. Then they would almost certainly want the "exact" value, so you'd give your answer as being simply "sqrt(3)".

Simplifying Square-Root Terms

To simplify a square root, you "take out" anything that is a "perfect square"; that is, you take out front anything that has two copies of the same factor:

    sqrt[4] = sqrt[2^2] = 2

    sqrt[49] = sqrt[7^2] = 7

    sqrt[225] = sqrt[15^2] = 15

Note that the value of the simplified radical is positive. While either of +2 and –2 might have been squared to get 4, "the square root of four" is defined to be only the positive option, +2. When you solve the equation x2 = 4, you are trying to find all possible values that might have been squared to get 4. But when you are just simplifying the expression sqrt[4], the ONLY answer is "2"; this positive result is called the "principal" root. (Other roots, such as –2, can be defined using graduate-school topics like "complex analysis" and "branch functions", but you won't need that for years, if ever.)

Sometimes the argument of a radical is not a perfect square, but it may "contain" a square amongst its factors. To simplify, you need to factor the argument and "take out" anything that is a square; you find anything you've got a pair of inside the radical, and you move it out front. To do this, you use the fact that you can switch between the multiplication of roots and the root of a multiplication. In other words, radicals can be manipulated similarly to powers:

    (ab)^n = a^n b^n, and the n-th root of (ab) equals the n-th root of a times the n-th root of b

  • Simplify sqrt[144]
  • There are various ways I can approach this simplification. One would be by factoring and then taking two different square roots:

      sqrt[144] = sqrt[9 * 16] = sqrt[9] sqrt[16] = 3 * 4 = 12

    The square root of 144 is 12.

You probably already knew that 122 = 144, so obviously the square root of 144 must be 12. But my steps above show how you can switch back and forth between the different formats (multiplication inside one radical, versus multiplication of two radicals) to help in the simplification process.

  • Simplify sqrt[24] sqrt[6]
  • Neither of 24 and 6 is a square, but what happens if I multiply them inside one radical?

      sqrt[24] sqrt[6] = sqrt[24 * 6] = sqrt[144] = sqrt[12 * 12] = 12

  • Simplify sqrt[75]
    • sqrt[75] = sqrt[3 * 25] = sqrt[25] sqrt[3] = 5 sqrt[3]

This answer is pronounced as "five, root three". It is proper form to put the radical at the end of the expression. Not only is "sqrt[3]5" non-standard, it is very hard to read, especially when hand-written. And write neatly, because "5 sqrt[3], or 'five times the square root of three'" is not the same as "5th-rt[3], or 'the fifth root of three'".

You don't have to factor the radicand all the way down to prime numbers when simplifying. As soon as you see a pair of factors or a perfect square, you've gone far enough.

  • Simplify sqrt[72]
  • Since 72 factors as 2×36, and since 36 is a perfect square, then:

      sqrt[72] = sqrt[2 * 36] = sqrt[2 * (6 * 6)] = 6 sqrt[2]

Since there had been only one copy of the factor 2 in the factorization 2×6×6, that left-over 2 couldn't come out of the radical and had to be left behind.

  • Simplify sqrt[4500]
    • sqrt[4500] = sqrt[45 * 100] = sqrt[5 * 9 * 100] = 3 * 10 * sqrt[5] = 30 sqrt[5] 

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Cite this article as:

Stapel, Elizabeth. "Square Roots: Introduction and Simplification." Purplemath. Available from Accessed



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