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Math 8 · 8.EE.A.2

Square roots

You know the area of a square. How long is its side?

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When you reach for it

Squaring takes a number and multiplies it by itself: 6^2 = 6 \times 6 = 36. Many questions run the other way. You are told the result, 36, and asked what number was squared to give it.

The plainest case is a square whose area you know. Thirty-six square tiles can be laid out as a square, and the question “how many tiles along each side?” asks for the number whose square is 36. The answer is 6 (Figure 1).

6
6
Figure 136 tiles make a square with 6 along each side, because 6^2 = 36. So \displaystyle \sqrt{36} = 6.

Undoing a square is called taking a square root. This lesson takes square roots of the numbers whose roots can be read straight off a square you already know: the perfect squares 1, 4, 9, 16, 25, \dots, and fractions and decimals built from them, such as \displaystyle \frac{25}{36} and 0.64. For a number like 50, which is not the square of a whole number, see Approximating irrational numbers.

The procedure

Two numbers square to the same thing

A square root of a number a is a number whose square is a.

Both 5^2 = 25 and (-5)^2 = (-5)(-5) = 25, because the product of two negative numbers is positive. So 25 has two square roots, 5 and -5. The same happens for every positive number: if r is a square root of a, then so is -r, since (-r)^2 = r^2. A positive number has exactly two square roots, one positive and one negative, and they are opposites.

Zero has only one square root, 0, because 0^2 = 0 and every other number squares to something positive.

A negative number has no square root among the real numbers. A positive number squared is positive, a negative number squared is positive, and 0 squared is 0, so no real number squared gives -25.

The radical sign names one number

The symbol \displaystyle \sqrt{\phantom{a}} is called the radical sign, and the number written under it is the radicand. The radical sign means the square root that is not negative, called the principal square root:

\sqrt{a} \text{ is the number that is } \ge 0 \text{ and whose square is } a.

So \displaystyle \sqrt{25} = 5, and not -5. The reason for choosing one is that a symbol in an expression has to stand for a single number. If \displaystyle \sqrt{25} could be either 5 or -5, then \displaystyle 2\sqrt{25} would be either 10 or -10, and no calculation with it could be finished.

The negative root is written with a minus sign in front of the radical, \displaystyle -\sqrt{25} = -5. When both roots are wanted together, they are written \displaystyle \pm\sqrt{25}, read “plus or minus the square root of 25”, which is shorthand for the pair 5 and -5. Zero fits the definition too: \displaystyle \sqrt{0} = 0.

Two facts follow straight from the definition, for every a \ge 0:

\sqrt{a} \ge 0 \qquad\text{and}\qquad \left(\sqrt{a}\right)^2 = a.

The second one is how every answer in this lesson is checked: square it and see whether the radicand comes back.

Reading a root off a known square

If n \ge 0, then \displaystyle \sqrt{n^2} = n, since n is not negative and its square is n^2. So the square root of a perfect square is read straight off a list of squares. These are worth knowing by heart, in both directions:

nn^2nn^2nn^2
1163611121
2474912144
3986413169
41698114196
5251010015225

To find \displaystyle \sqrt{196}, find 196 in a square column: it sits beside 14, so \displaystyle \sqrt{196} = 14.

Fractions and decimals

A fraction is squared by squaring its top and its bottom: \displaystyle \left(\frac{p}{q}\right)^2 = \frac{p^2}{q^2}. Read backwards, when the top and bottom of a fraction are both perfect squares, its square root is the root of the top over the root of the bottom. For example \displaystyle \left(\frac{3}{4}\right)^2 = \frac{9}{16}, so

\sqrt{\frac{9}{16}} = \frac{3}{4}.

A decimal is a fraction with a denominator of 10, 100, 1000 and so on, so the same idea applies. \displaystyle 0.64 = \frac{64}{100}, and both 64 and 100 are perfect squares, so

\sqrt{0.64} = \frac{8}{10} = 0.8.

Figure 2 shows the same fact as an area: a 1 by 1 square cut into 100 small squares, of which an 8 by 8 block is shaded. The block covers 64 hundredths of the whole, and its side is 8 tenths.

0.8
0.8
1
Figure 2The shaded square covers 64 of the 100 small squares, so its area is 0.64, and its side is 0.8. So \displaystyle \sqrt{0.64} = 0.8.

Squaring a decimal doubles the number of decimal places, because tenths times tenths gives hundredths, and hundredths times hundredths gives ten-thousandths. Read backwards, when the root of a decimal with 2 places is a terminating decimal it has 1 place, and with 4 places it has 2.

An equation asks for every root, the symbol names one

The equation x^2 = 25 asks for every number whose square is 25. There are two, so

x^2 = 25 \quad\text{means}\quad x = 5 \text{ or } x = -5, \text{ written } x = \pm 5.

The expression \displaystyle \sqrt{25} is not a question; it names one number, 5. In general the solutions of x^2 = a, for a > 0, are \displaystyle \sqrt{a} and \displaystyle -\sqrt{a}. Solving such equations when the square is of a bracket or the answer is not a whole number is solving quadratics with square roots.

The side of a square from its area

A square of side s has area s^2, so if the area A is known, the side solves s^2 = A. Of the two solutions only the one that is not negative can be a length, so

s = \sqrt{A}.

Finding a circle’s radius from its area works the same way once the area is divided by \pi; see Area of a circle.

Worked examples

Example 1

Whole-number radicands

Evaluate each expression, or explain why it is not a real number.

(a) \displaystyle \sqrt{169} \qquad (b) \displaystyle -\sqrt{576} \qquad (c) \displaystyle \sqrt{-49} \qquad (d) \displaystyle \sqrt{0}

Solution

(a) 13^2 = 169, and 13 is not negative, so \displaystyle \sqrt{169} = 13.

(b) The minus sign sits outside the radical, so the root is taken first and the result is then made negative. 24^2 = 576, so \displaystyle \sqrt{576} = 24 and \displaystyle -\sqrt{576} = -24.

(c) Here the minus sign is under the radical, so the question is which number squares to -49. No real number does: a positive or a negative number squared is positive, and 0 squared is 0. So \displaystyle \sqrt{-49} is not a real number. Compare (b): \displaystyle -\sqrt{49} = -7 is a perfectly good number, because there the root is taken of 49, not of -49.

(d) 0^2 = 0, so \displaystyle \sqrt{0} = 0.

Example 2

Fractions, decimals and a mixed number

Evaluate (a) \displaystyle \sqrt{\frac{25}{36}} \qquad (b) \displaystyle \sqrt{0.64} \qquad (c) \displaystyle \sqrt{0.0004} \qquad (d) \displaystyle \sqrt{1\frac{9}{16}}

Solution

(a) Both 25 and 36 are perfect squares, 5^2 and 6^2, and \displaystyle \left(\frac{5}{6}\right)^2 = \frac{25}{36}. So \displaystyle \sqrt{\frac{25}{36}} = \frac{5}{6}.

(b) Write the decimal as a fraction: \displaystyle 0.64 = \frac{64}{100}. The root of the top is 8 and of the bottom is 10, so \displaystyle \sqrt{0.64} = 0.8. Check: 0.8 \times 0.8 = 0.64.

(c) 0.0004 has four decimal places, so it is \displaystyle \frac{4}{10000}, and \displaystyle \sqrt{\frac{4}{10000}} = \frac{2}{100}. So \displaystyle \sqrt{0.0004} = 0.02. The root has half as many decimal places, two. Check: 0.02 \times 0.02 = 0.0004.

(d) A mixed number is a sum, \displaystyle 1\frac{9}{16} = 1 + \frac{9}{16}, and the radical covers the whole sum. Write it as one fraction:

1\frac{9}{16} = \frac{16}{16} + \frac{9}{16} = \frac{25}{16}.

Now \displaystyle \sqrt{25} = 5 and \displaystyle \sqrt{16} = 4, so \displaystyle \sqrt{1\frac{9}{16}} = \frac{5}{4} = 1\frac{1}{4}. Check: \displaystyle \left(\frac{5}{4}\right)^2 = \frac{25}{16} = 1\frac{9}{16}. The two parts do not root separately: \displaystyle \sqrt{1} + \sqrt{\frac{9}{16}} = 1\frac{3}{4} = \frac{7}{4}, whose square is \displaystyle \frac{49}{16}, not \displaystyle \frac{25}{16}.

Example 3

The equation and the symbol

(a) Solve x^2 = 100. \qquad (b) Evaluate \displaystyle \sqrt{100}. \qquad (c) Solve x^2 = 0.16. \qquad (d) Solve x^2 = -81.

Solution

(a) The equation asks for every number whose square is 100. Both 10^2 = 100 and (-10)^2 = 100, so x = \pm 10: two solutions.

(b) The radical sign names the root that is not negative, so \displaystyle \sqrt{100} = 10: one number.

(c) \displaystyle 0.16 = \frac{16}{100}, and \displaystyle \sqrt{0.16} = \frac{4}{10} = 0.4. The equation wants both roots, so x = \pm 0.4. Check: (-0.4)^2 = 0.16 as well as 0.4^2 = 0.16.

(d) No real number squared is negative, so the equation has no real solution.

Example 4

Where the radical sign groups

Evaluate each pair and compare.

(a) \displaystyle \sqrt{36 + 64} and \displaystyle \sqrt{36} + \sqrt{64} \qquad (b) \displaystyle \sqrt{100 - 36} and \displaystyle \sqrt{100} - \sqrt{36} \qquad (c) \displaystyle \sqrt{(-9)^2}

Solution

The bar of the radical sign works like brackets: everything under it is worked out first, and only then is the root taken. This is part of the Order of operations.

(a) Under the bar, 36 + 64 = 100, so \displaystyle \sqrt{36 + 64} = \sqrt{100} = 10. The other expression takes two roots and then adds: \displaystyle \sqrt{36} + \sqrt{64} = 6 + 8 = 14. The answers differ, so the root of a sum is not the sum of the roots.

(b) Under the bar, 100 - 36 = 64, so \displaystyle \sqrt{100 - 36} = \sqrt{64} = 8. Separately, \displaystyle \sqrt{100} - \sqrt{36} = 10 - 6 = 4. Again different.

(c) The square is under the bar, so it comes first: (-9)^2 = 81. Then \displaystyle \sqrt{81} = 9. So \displaystyle \sqrt{(-9)^2} = 9. The root of a square gives back the number only when that number is not negative; for -9 it gives back its opposite.

Example 5

A square garden

Henrike has a square garden with an area of 3.24 square meters. How long is each side, and how much fencing does she need to go all the way round it?

Solution

Call the side s meters. The area of a square is its side squared, so

s^2 = 3.24.

This equation has two solutions, \displaystyle \sqrt{3.24} and \displaystyle -\sqrt{3.24}, but a side is a length and cannot be negative, so \displaystyle s = \sqrt{3.24}. Write the decimal as a fraction: \displaystyle 3.24 = \frac{324}{100}. Since 18^2 = 324, \displaystyle \sqrt{324} = 18, and

s = \frac{18}{10} = 1.8.

Each side is 1.8 meters. Check: 1.8 \times 1.8 = 3.24. The fence runs along four sides, so it is 4 \times 1.8 = 7.2 meters long.

Example 6

The largest square from a pile of tiles

Jorunn has 150 square tiles, all the same size, and wants to lay as many as she can in one solid square. How many tiles go along each side, and how many are left over? How many more tiles would she need for the next size of square?

Solution

A solid square with n tiles along each side uses n^2 tiles. So the question is which is the largest perfect square not bigger than 150. From the list of squares, 12^2 = 144 and 13^2 = 169. Since 144 \le 150 < 169, the largest square she can make is 12 tiles along each side, using 144 tiles, with 150 - 144 = 6 tiles left over.

The next size, 13 along each side, needs 169 tiles, which is 169 - 150 = 19 more than she has.

The same comparison says that \displaystyle \sqrt{150} lies between 12 and 13: 150 is not a perfect square, so its root is not a whole number. Pinning such a root down more closely is Approximating irrational numbers.

Boundaries

  • The radicand cannot be negative, among the real numbers, since no real number squares to a negative number. A larger number system later gives such roots a meaning; see Imaginary and complex numbers. A minus sign outside the radical is fine: \displaystyle -\sqrt{49} = -7.
  • \displaystyle \sqrt{a^2} = a only when a \ge 0. For a negative number the root of its square is its opposite: \displaystyle \sqrt{(-9)^2} = 9.
  • The radical does not split over addition or subtraction, as Example 4 shows. It does split over multiplication and division of numbers that are not negative, which is what made \displaystyle \sqrt{\frac{25}{36}} = \frac{5}{6} work, and which Simplifying a square root uses on radicands that are not perfect squares.
  • Reading off only works for perfect squares and for fractions and decimals built from them. A decimal with an odd number of decimal places, such as 0.9, is never the square of a terminating decimal (Exercise 25). The roots of 50 or 0.9 are not fractions at all; see Irrational numbers.
  • A length takes only the root that is not negative.

Where this goes next

The roots of numbers that are not perfect squares sit between the roots you can read off, so \displaystyle \sqrt{150} is between 12 and 13, and Approximating irrational numbers narrows them down. Most of them cannot be written as a fraction at all, the subject of Irrational numbers.

The same question one power up asks which number cubed gives 125; that is Cube roots and higher roots, where a negative number does have a root, since (-5)^3 = -125.

The side-from-area idea runs through geometry: the The Pythagorean theorem gives the square of a side of a right triangle, and a square root turns it back into a length. In algebra, the pair of roots behind x^2 = 25 grows into solving quadratics with square roots, and radicands that are not perfect squares are tidied up in Simplifying a square root.

Exercises

Square roots of whole numbers

Evaluate each expression, or explain why it is not a real number.

  1. \displaystyle \sqrt{121}
  2. \displaystyle \sqrt{225}
  3. \displaystyle -\sqrt{144}
  4. \displaystyle \sqrt{1}
  5. \displaystyle \sqrt{-100}

Fractions and decimals

Evaluate.

  1. \displaystyle \sqrt{\frac{4}{49}}
  2. \displaystyle \sqrt{\frac{81}{100}}
  3. \displaystyle \sqrt{0.25}
  4. \displaystyle \sqrt{0.09}
  5. \displaystyle \sqrt{1.21}
  6. \displaystyle \sqrt{2\frac{7}{9}}

The equation or the symbol

Solve each equation, giving every real solution.

  1. x^2 = 900
  2. \displaystyle y^2 = \frac{1}{16}
  3. m^2 = 0
  4. x^2 = -400

Order of operations

Evaluate.

  1. \displaystyle \sqrt{144 - 44}
  2. \displaystyle \sqrt{9^2 + 12^2}
  3. \displaystyle 5\sqrt{36} - \sqrt{4 \cdot 9}
  4. \displaystyle \sqrt{(-12)^2}

Problems

  1. A square playground has an area of 361 square meters. How long is each side, and how much fencing goes all the way round it?
  2. Kaito covers a square floor of area 7.29 square meters with square tiles 0.3 meters on each side, with no gaps and no cut tiles. How long is each side of the floor, how many tiles fit along one side, and how many tiles does he use?
  3. Lucinda mounts a square photo of area 225 square centimeters in the middle of a square card of area 400 square centimeters, leaving a border of the same width all the way round. How wide is the border?

Reasoning

  1. Explain why \displaystyle \sqrt{-16} is not a real number, while \displaystyle -\sqrt{16} is. What is \displaystyle -\sqrt{16}?
  2. Ibrahim-Ali solves x^2 = 81 and writes x = \pm 9. On the next line he evaluates \displaystyle \sqrt{81} and writes 9. Explain why the first answer has two numbers and the second has only one, and why both are right.
  3. A terminating decimal has k decimal places and its last digit is not 0. Show that its square has exactly 2k decimal places. Use this to explain why 0.9 is not the square of any terminating decimal, even though 0.09 is.
Answers
  1. 11. 11^2 = 121.
  2. 15. 15^2 = 225.
  3. -12. \displaystyle \sqrt{144} = 12, and the minus sign outside the radical makes it -12.
  4. 1. 1^2 = 1, and 1 is not negative.
  5. Not a real number. No real number squared is negative, so nothing squares to -100.
  6. \displaystyle \frac{2}{7}. 2^2 = 4 and 7^2 = 49.
  7. \displaystyle \frac{9}{10}, or 0.9. 9^2 = 81 and 10^2 = 100.
  8. 0.5. \displaystyle 0.25 = \frac{25}{100}, and \displaystyle \sqrt{25} = 5, \displaystyle \sqrt{100} = 10.
  9. 0.3. \displaystyle 0.09 = \frac{9}{100}, so the root is \displaystyle \frac{3}{10}. Check: 0.3 \times 0.3 = 0.09.
  10. 1.1. \displaystyle 1.21 = \frac{121}{100}, so the root is \displaystyle \frac{11}{10}. Check: 1.1 \times 1.1 = 1.21.
  11. \displaystyle \frac{5}{3} = 1\frac{2}{3}. \displaystyle 2\frac{7}{9} = \frac{18}{9} + \frac{7}{9} = \frac{25}{9}, and \displaystyle \sqrt{25} = 5, \displaystyle \sqrt{9} = 3.
  12. x = \pm 30. Both 30^2 and (-30)^2 are 900.
  13. \displaystyle y = \pm\frac{1}{4}. \displaystyle \left(\frac{1}{4}\right)^2 = \frac{1}{16}, and so is \displaystyle \left(-\frac{1}{4}\right)^2.
  14. m = 0, the only solution. Every other number squares to something positive.
  15. No real solution. A real number squared is never negative.
  16. 10. Work under the bar first: 144 - 44 = 100, and \displaystyle \sqrt{100} = 10.
  17. 15. Under the bar: 81 + 144 = 225, and \displaystyle \sqrt{225} = 15. This is not \displaystyle \sqrt{9^2} + \sqrt{12^2} = 9 + 12 = 21.
  18. 24. \displaystyle 5\sqrt{36} = 5 \times 6 = 30, and \displaystyle \sqrt{4 \cdot 9} = \sqrt{36} = 6, so 30 - 6 = 24.
  19. 12. (-12)^2 = 144 first, then \displaystyle \sqrt{144} = 12.
  20. Each side is 19 meters and the fencing is 76 meters. The side solves s^2 = 361 with s > 0, so \displaystyle s = \sqrt{361} = 19, and 4 \times 19 = 76.
  21. Each side is 2.7 meters, 9 tiles fit along a side, and he uses 81 tiles. \displaystyle \sqrt{7.29} = \sqrt{\frac{729}{100}} = \frac{27}{10}, since 27^2 = 729. Then 2.7 \div 0.3 = 9, and a 9 by 9 square of tiles has 81. Check: each tile covers 0.09 square meters, and 81 \times 0.09 = 7.29.
  22. 2.5 centimeters. The card has side \displaystyle \sqrt{400} = 20 and the photo has side \displaystyle \sqrt{225} = 15. The 5 centimeters left over across the card is shared between two borders, one on each side, so each is 5 \div 2 = 2.5.
  23. \displaystyle -\sqrt{16} = -4. In \displaystyle \sqrt{-16} the root is asked of -16, and no real number squares to a negative number. In \displaystyle -\sqrt{16} the root is taken of 16, which is 4, and only then is the minus sign applied.
  24. The equation x^2 = 81 asks for every number whose square is 81, and there are two, 9 and -9. The expression \displaystyle \sqrt{81} is the name of one number, the square root that is not negative, which is 9. They answer different questions, and they agree: the solutions of x^2 = 81 are \displaystyle \sqrt{81} and \displaystyle -\sqrt{81}.
  25. Write the decimal as \displaystyle \frac{n}{10^k}, where n is a whole number whose last digit d is not 0. Its square is \displaystyle \frac{n^2}{10^{2k}}. The last digit of n^2 is the last digit of d^2, and the squares of 1 to 9 are 1, 4, 9, 16, 25, 36, 49, 64, 81, none ending in 0. So n^2 does not end in 0, nothing cancels, and the square has exactly 2k decimal places, an even number. 0.9 has one decimal place, an odd number, so it is not such a square. 0.09 has two, and it is 0.3^2. (The square root of 0.9 still exists; it just does not terminate. See Irrational numbers.)
kg1128 · updated 2026-10-09 · CC-BY-SA-4.0