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Square Roots and Irrational Numbers

Simplify square roots by factoring

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Square Roots and Irrational Numbers

Square Roots and Irrational Numbers 

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*}a=b2, we say that \begin{align*}b\end{align*}b is the square root of \begin{align*}a\end{align*}a.

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*}x is written as \begin{align*}\sqrt{x}\end{align*}x or sometimes as \begin{align*}\sqrt[2]{x}\end{align*}x2. 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.

License: CC BY-NC 3.0

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.

 

Finding the Principal Square Root 

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

a) 121

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

b) 225

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

c) 324

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

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*}ab, where \begin{align*}a\end{align*}a is the product of half the paired factors we pulled out and \begin{align*}b\end{align*}b is the product of the leftover factors.

Finding the Principal Square Roots of Imperfect Squares 

Find the principal square root of the following numbers.

a) 8

\begin{align*}8 = 2 \times 2 \times 2\end{align*}8=2×2×2. This gives us one pairs of 2's and one leftover 2, so \begin{align*}\sqrt{8} = 2 \sqrt{2}\end{align*}8=22.

b) 48

\begin{align*}48 = (2 \times 2) \times (2 \times 2) \times 3\end{align*}48=(2×2)×(2×2)×3, so \begin{align*}\sqrt{48} = 2 \times 2 \times \sqrt{3}\end{align*}48=2×2×3, or \begin{align*}4 \sqrt{3}\end{align*}43.

c) 75

\begin{align*}75 = (5 \times 5) \times 3\end{align*}75=(5×5)×3, so \begin{align*}\sqrt{75} = 5 \sqrt{3}\end{align*}75=53.

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*}a×b=ab
  • \begin{align*}A \sqrt{a} \times B \sqrt{b} = AB \sqrt{ab}\end{align*}Aa×Bb=ABab
  • \begin{align*}\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\end{align*}ab=ab
  • \begin{align*}\frac{A \sqrt{a}}{B \sqrt{b}} = \frac{A}{B} \sqrt{\frac{a}{b}}\end{align*}AaBb=ABab

Simplifying Square Roots 

Simplify the following square root problems

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

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

b) \begin{align*}3 \sqrt{4} \times 4 \sqrt{3}\end{align*}34×43

\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*}34×43=1212=12(2×2)×3=12×23=243

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

\begin{align*}\sqrt{12} \ \div \sqrt{3} = \sqrt{\frac{12}{3}} = \sqrt{4} = 2\end{align*}12 ÷3=123=4=2

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

\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.

Using a Calculator 

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

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

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

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

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

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

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

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

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

 

Examples 

Find the square root of each number.

Example 1

576

\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*} 

Example 2

216

\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*}.

Review 

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*}

Review (Answers)

To view the Review answers, open this PDF file and look for section 2.9. 

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Vocabulary

approximate solution

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

Irrational Number

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

Perfect Square

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

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

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

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.

Image Attributions

  1. [1]^ License: CC BY-NC 3.0

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