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Square Roots and Irrational Numbers

Simplify square roots by factoring

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Square Roots and Irrational Numbers

Suppose an elementary school has a square playground with an area of 3000 square feet. Could you find the width of the playground? Would the width be a rational or irrational number? In this Concept, you'll learn how to take the square root of a number and decide whether the result is rational or irrational so that you can answer questions such as these.

Guidance

Human chess is a variation of chess, often played at Renaissance fairs, in which people take on the roles of the various pieces on a chessboard. The chessboard is played on a square plot of land that measures 324 square meters with the chess squares marked on the grass. How long is each side of the chessboard?

To answer this question, you will need to know how to find the square root of a number.

The square root of a number n\begin{align*}n\end{align*} is any number s\begin{align*}s\end{align*} such that s2=n\begin{align*}s^2 = n\end{align*}.

Every positive number has two square roots, the positive and the negative. The symbol used to represent the square root is x\begin{align*}\sqrt{x}\end{align*}. It is assumed that this is the positive square root of x\begin{align*}x\end{align*}. To show both the positive and negative values, you can use the symbol ±\begin{align*}\pm\end{align*}, read “plus or minus.”

For example:

81=9\begin{align*}\sqrt{81}=9\end{align*} means the positive square root of 81.

81=9\begin{align*}-\sqrt{81}= -9\end{align*} means the negative square root of 81.

±81=±9\begin{align*}\pm \sqrt{81} = \pm 9\end{align*} means the positive or negative square root of 81.

Example A

The human chessboard measures 324 square meters. How long is one side of the square?

Solution: The area of a square is s2=Area\begin{align*}s^2 = Area\end{align*}. The value of Area can be replaced with 324.

s2=324\begin{align*}s^2=324\end{align*}

The value of s\begin{align*}s\end{align*} represents the square root of 324.

s=324=18\begin{align*}s= \sqrt{324}=18\end{align*}

The chessboard is 18 meters long by 18 meters wide.

Approximating Square Roots

When the square root of a number is a whole number, this number is called a perfect square. 9 is a perfect square because 9=3\begin{align*}\sqrt{9}=3\end{align*}.

Not all square roots are whole numbers. Many square roots are irrational numbers, meaning there is no rational number equivalent. For example, 2 is the square root of 4 because 2×2=4\begin{align*}2 \times 2 = 4\end{align*}. The number 7 is the square root of 49 because 7×7=49\begin{align*}7 \times 7 = 49\end{align*}. What is the square root of 5?

There is no whole number multiplied by itself that equals five, so 5\begin{align*}\sqrt{5}\end{align*} is not a whole number. To find the value of 5\begin{align*}\sqrt{5}\end{align*}, we can use estimation.

To estimate the square root of a number, look for the perfect integers less than and greater than the value, and then estimate the decimal.

Example B

Estimate 5\begin{align*}\sqrt{5}\end{align*}.

Solution: The perfect square below 5 is 4 and the perfect square above 5 is 9. Therefore, 4<5<9\begin{align*}4<5<9\end{align*}. Therefore, 5\begin{align*}\sqrt{5}\end{align*} is between 4\begin{align*}\sqrt{4}\end{align*} and 9\begin{align*}\sqrt{9}\end{align*}, or 2<5<3\begin{align*}2< \sqrt{5}<3\end{align*}. Because 5 is closer to 4 than 9, the decimal is a low value: 52.2\begin{align*}\sqrt{5} \approx 2.2\end{align*}.

Identifying Irrational Numbers

Recall the number hierarchy from a previous Concept. Real numbers have two categories: rational and irrational. If a value is not a perfect square, then it is considered an irrational number. These numbers cannot be written as a fraction because the decimal does not end (non-terminating) and does not repeat a pattern (non-repeating). Although irrational square roots cannot be written as fractions, we can still write them exactly, without typing the value into a calculator.

For example, suppose you do not have a calculator and you need to find 18\begin{align*}\sqrt{18}\end{align*}. You know there is no whole number squared that equals 18, so 18\begin{align*}\sqrt{18}\end{align*} is an irrational number. The value is between 16=4\begin{align*}\sqrt{16}=4\end{align*} and 25=5\begin{align*}\sqrt{25}=5\end{align*}. However, we need to find the exact value of 18\begin{align*}\sqrt{18}\end{align*}.

Begin by writing the prime factorization of 18\begin{align*}\sqrt{18}\end{align*}. 18=9×2=9×2\begin{align*}\sqrt{18} = \sqrt{9 \times 2}= \sqrt{9} \times \sqrt{2}\end{align*}. 9=3\begin{align*}\sqrt{9} = 3\end{align*} but 2\begin{align*}\sqrt{2}\end{align*} does not have a whole number value. Therefore, the exact value of 18\begin{align*}\sqrt{18}\end{align*} is 32\begin{align*}3 \sqrt{2}\end{align*}.

You can check your answer on a calculator by finding the decimal approximation for each square root.

Example C

Find the exact value of 75\begin{align*}\sqrt{75}\end{align*}.

Solution:

75=25×3=25×3=53\begin{align*}\sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5 \sqrt{3}\end{align*}

Guided Practice

The area of a square is 50 square feet. What are the lengths of its sides?

Solution:

We know that the formula for the area of a square is a=s2.\begin{align*}a=s^2.\end{align*} Using this formula:

a=s250=s250=s250=s\begin{align*}&a=s^2\\ &50=s^2\\ &\sqrt{50}=\sqrt{s^2}\\ &\sqrt{50}=s\end{align*}

Now we will simplify:

50=252=52.\begin{align*}\sqrt{50}=\sqrt{25\cdot 2}=5\sqrt{2}.\end{align*}

The length of each side of the square is 52\begin{align*}5\sqrt{2}\end{align*} feet.

Practice

Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: Square Roots and Real Numbers (10:18)

Find the following square roots exactly without using a calculator. Give your answer in the simplest form.

1. 25\begin{align*}\sqrt{25}\end{align*}
2. 24\begin{align*}\sqrt{24}\end{align*}
3. 20\begin{align*}\sqrt{20}\end{align*}
4. 200\begin{align*}\sqrt{200}\end{align*}
5. 2000\begin{align*}\sqrt{2000}\end{align*}
6. 14\begin{align*}\sqrt{\frac{1}{4}}\end{align*}
7. 94\begin{align*}\sqrt{\frac{9}{4}}\end{align*}
8. 0.16\begin{align*}\sqrt{0.16}\end{align*}
9. \begin{align*}\sqrt{0.1}\end{align*}
10. \begin{align*}\sqrt{0.01}\end{align*}

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{0.25}\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*}

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