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2.9: Square Roots and Irrational Numbers

Difficulty Level: At Grade Created by: CK-12
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What if you had a number like 1000 and you wanted to find its square root? After completing this concept, you'll be able to find square roots like this one by hand and with a calculator.

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CK-12 Foundation: 0209S Square Roots (H264)

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You can also work out square roots by hand using a method similar to long division. (See the web page at http://www.homeschoolmath.net/teaching/square-root-algorithm.php for an explanation of this method.)


The square root of a number is a number which, when multiplied by itself, gives the original number. In other words, if \begin{align*}a = b^2\end{align*}, we say that \begin{align*}b\end{align*} is the square root of \begin{align*}a\end{align*}.

Note: Negative numbers and positive numbers both yield positive numbers when squared, so each positive number has both a positive and a negative square root. (For example, 3 and -3 can both be squared to yield 9.) The positive square root of a number is called the principal square root.

The square root of a number \begin{align*}x\end{align*} is written as \begin{align*}\sqrt{x}\end{align*} or sometimes as \begin{align*}\sqrt[2]{x}\end{align*}. The symbol \begin{align*}\sqrt{\;\;}\end{align*} is sometimes called a radical sign.

Numbers with whole-number square roots are called perfect squares. The first five perfect squares (1, 4, 9, 16, and 25) are shown below.

You can determine whether a number is a perfect square by looking at its prime factors. If every number in the factor tree appears an even number of times, the number is a perfect square. To find the square root of that number, simply take one of each pair of matching factors and multiply them together.

Example A

Find the principal square root of each of these perfect squares.

a) 121

b) 225

c) 324


a) \begin{align*}121 = 11 \times 11\end{align*}, so \begin{align*}\sqrt{121} = 11\end{align*}.

b) \begin{align*}225 = (5 \times 5) \times (3 \times 3)\end{align*}, so \begin{align*}\sqrt{225} = 5 \times 3 = 15\end{align*}.

c) \begin{align*}324 = (2 \times 2) \times (3 \times 3) \times (3 \times 3)\end{align*}, so \begin{align*}\sqrt{324} = 2 \times 3 \times 3 = 18\end{align*}.

For more practice matching numbers with their square roots, try the Flash games at http://www.quia.com/jg/65631.html.

When the prime factors don’t pair up neatly, we “factor out” the ones that do pair up and leave the rest under a radical sign. We write the answer as \begin{align*}a \sqrt{b}\end{align*}, where \begin{align*}a\end{align*} is the product of half the paired factors we pulled out and \begin{align*}b\end{align*} is the product of the leftover factors.

Example B

Find the principal square root of the following numbers.

a) 8

b) 48

c) 75


a) \begin{align*}8 = 2 \times 2 \times 2\end{align*}. This gives us one pair of 2’s and one leftover 2, so \begin{align*}\sqrt{8} = 2 \sqrt{2}\end{align*}.

b) \begin{align*}48 = (2 \times 2) \times (2 \times 2) \times 3\end{align*}, so \begin{align*}\sqrt{48} = 2 \times 2 \times \sqrt{3}\end{align*}, or \begin{align*}4 \sqrt{3}\end{align*}.

c) \begin{align*}75 = (5 \times 5) \times 3\end{align*}, so \begin{align*}\sqrt{75} = 5 \sqrt{3}\end{align*}.

Note that in the last example we collected the paired factors first, then we collected the unpaired ones under a single radical symbol. Here are the four rules that govern how we treat square roots.

  • \begin{align*}\sqrt{a} \times \sqrt{b} = \sqrt{ab}\end{align*}
  • \begin{align*}A \sqrt{a} \times B \sqrt{b} = AB \sqrt{ab}\end{align*}
  • \begin{align*}\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\end{align*}
  • \begin{align*}\frac{A \sqrt{a}}{B \sqrt{b}} = \frac{A}{B} \sqrt{\frac{a}{b}}\end{align*}

Example C

Simplify the following square root problems

a) \begin{align*}\sqrt{8} \times \sqrt{2}\end{align*}

b) \begin{align*}3 \sqrt{4} \times 4 \sqrt{3}\end{align*}

c) \begin{align*}\sqrt{12} \ \div \sqrt{3}\end{align*}

d) \begin{align*}12 \sqrt{10} \div 6 \sqrt{5}\end{align*}


a) \begin{align*}\sqrt{8} \times \sqrt{2} = \sqrt{16} = 4\end{align*}

b) \begin{align*}3 \sqrt{4} \times 4 \sqrt{3} = 12 \sqrt{12} = 12 \sqrt{(2 \times 2) \times 3} = 12 \times 2 \sqrt{3} = 24 \sqrt{3}\end{align*}

c) \begin{align*}\sqrt{12} \ \div \sqrt{3} = \sqrt{\frac{12}{3}} = \sqrt{4} = 2\end{align*}

d) \begin{align*}12 \sqrt{10} \div 6 \sqrt{5} = \frac{12}{6} \sqrt{\frac{10}{5}} = 2 \sqrt{2}\end{align*}

Approximate Square Roots

Terms like \begin{align*}\sqrt{2}, \sqrt{3}\end{align*} and \begin{align*}\sqrt{7}\end{align*} (square roots of prime numbers) cannot be written as rational numbers. That is to say, they cannot be expressed as the ratio of two integers. We call them irrational numbers. In decimal form, they have an unending, seemingly random, string of numbers after the decimal point.

To find approximate values for square roots, we use the \begin{align*}\sqrt{\;\;}\end{align*} or \begin{align*}\sqrt{x}\end{align*} button on a calculator. When the number we plug in is a perfect square, or the square of a rational number, we will get an exact answer. When the number is a non-perfect square, the answer will be irrational and will look like a random string of digits. Since the calculator can only show some of the infinitely many digits that are actually in the answer, it is really showing us an approximate answer—not exactly the right answer, but as close as it can get.

Example D

Use a calculator to find the following square roots. Round your answer to three decimal places.

a) \begin{align*}\sqrt{99}\end{align*}

b) \begin{align*}\sqrt{5}\end{align*}

c) \begin{align*}\sqrt{0.5}\end{align*}

d) \begin{align*}\sqrt{1.75}\end{align*}


a) \begin{align*}\approx 9.950\end{align*}

b) \begin{align*}\approx 2.236\end{align*}

c) \begin{align*}\approx 0.707\end{align*}

d) \begin{align*}\approx 1.323\end{align*}

Watch this video for help with the Examples above.

CK-12 Foundation: Square Roots


  • The square root of a number is a number which gives the original number when multiplied by itself. In algebraic terms, for two numbers \begin{align*}a\end{align*} and \begin{align*}b\end{align*}, if \begin{align*}a = b^2\end{align*}, then \begin{align*}b = \sqrt{a}\end{align*}.
  • A square root can have two possible values: a positive value called the principal square root, and a negative value (the opposite of the positive value).
  • A perfect square is a number whose square root is an integer.
  • Some mathematical properties of square roots are:
    • \begin{align*}\sqrt{a} \ \times \sqrt{b} = \sqrt{ab}\end{align*}
    • \begin{align*}A \sqrt{a} \ \times B \sqrt{b} = AB \sqrt{ab}\end{align*}
    • \begin{align*}\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{ \frac{a}{b}}\end{align*}
    • \begin{align*}\frac{A\sqrt{a}}{B \sqrt{b}} = \frac{A}{B} \sqrt{\frac{a}{b}}\end{align*}
  • Square roots of numbers that are not perfect squares (or ratios of perfect squares) are irrational numbers. They cannot be written as rational numbers (the ratio of two integers). In decimal form, they have an unending, seemingly random, string of numbers after the decimal point.
  • Computing a square root on a calculator will produce an approximate solution since the calculator only shows a finite number of digits after the decimal point.

Guided Practice

Find the square root of each number.

a) 576

b) 216


a) \begin{align*}576 = (2 \times 2) \times (2 \times 2) \times (2 \times 2) \times (3 \times 3)\end{align*}, so \begin{align*}\sqrt{576} = 2 \times 2 \times 2 \times 3 = 24\end{align*}.

b) \begin{align*}216 = (2 \times 2) \times 2 \times (3 \times 3) \times 3\end{align*}, so \begin{align*}\sqrt{216} = 2 \times 3 \times \sqrt{2 \times 3}\end{align*}, or \begin{align*} 6 \sqrt{6}\end{align*}.


For 1-10, find the following square roots exactly without using a calculator, giving your answer in the simplest form.

  1. \begin{align*}\sqrt{25}\end{align*}
  2. \begin{align*}\sqrt{24}\end{align*}
  3. \begin{align*}\sqrt{20}\end{align*}
  4. \begin{align*}\sqrt{200}\end{align*}
  5. \begin{align*}\sqrt{2000}\end{align*}
  6. \begin{align*}\sqrt{\frac{1}{4}}\end{align*} (Hint: The division rules you learned can be applied backwards!)
  7. \begin{align*}\sqrt{\frac{9}{4}}\end{align*}
  8. \begin{align*}\sqrt{0.16}\end{align*}
  9. \begin{align*}\sqrt{0.1}\end{align*}
  10. \begin{align*}\sqrt{0.01}\end{align*}

For 11-20, use a calculator to find the following square roots. Round to two decimal places.

  1. \begin{align*}\sqrt{13}\end{align*}
  2. \begin{align*}\sqrt{99}\end{align*}
  3. \begin{align*}\sqrt{123}\end{align*}
  4. \begin{align*}\sqrt{2}\end{align*}
  5. \begin{align*}\sqrt{2000}\end{align*}
  6. \begin{align*}\sqrt{.25}\end{align*}
  7. \begin{align*}\sqrt{1.35}\end{align*}
  8. \begin{align*}\sqrt{0.37}\end{align*}
  9. \begin{align*}\sqrt{0.7}\end{align*}
  10. \begin{align*}\sqrt{0.01}\end{align*}


approximate solution

approximate solution

An approximate solution to a problem is a solution that has been rounded to a limited number of digits.
Irrational Number

Irrational Number

An irrational number is a number that can not be expressed exactly as the quotient of two integers.
Perfect Square

Perfect Square

A perfect square is a number whose square root is an integer.
principal square root

principal square root

The principal square root is the positive square root of a number, to distinguish it from the negative value. 3 is the principal square root of 9; -3 is also a square root of 9, but it is not principal square root.
rational number

rational number

A rational number is a number that can be expressed as the quotient of two integers, with the denominator not equal to zero.
Square Root

Square Root

The square root of a term is a value that must be multiplied by itself to equal the specified term. The square root of 9 is 3, since 3 * 3 = 9.

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Difficulty Level:
At Grade
Date Created:
Aug 13, 2012
Last Modified:
Apr 11, 2016
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