Squares and square roots

Section 1.5 Squares and square roots

Squares and square roots are fundamental concepts in mathematics, with many applications in our daily lives.

Consider a surveyor tasked with determining the side length of a square garden with an area of \(400\) m\(^2\text{.}\) Since all sides of a square are equal, the surveyor needs to find a number that, when multiplied by itself, results in \(400\text{.}\) This is where the concept of square roots comes into play.

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To find the side length of the garden, the surveyor needs to determine the square root of \(400\text{.}\) In this case, the square root of \(400\) is \(20\) since \(20 \times 20 = 400\text{,}\) meaning each side of the garden is \(20\) m long.

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We can represent this mathematically as follows:

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A square with side length a — Figure 1.5.1. A square garden
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Area of a square = side length \(\times\) side length
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\(a^2 = a \times a\) , where \(a\) represents the side length of the square.
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Subsection 1.5.1 Squares of whole numbers

Whole numbers are set of numbers including \(0\) and positive integers. They don’t contain decimals, fraction or negative integers. Whole numbers are also called counting numbers. i.e \(0,1,2,3,4,5,6,...\)

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We find the square of a whole number, by multiplying the whole number by itself, the resulting product is called the square number. In particular, the operation of multiplying a whole number with itself can be expressed with an exponent of 2.

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For example, the square of \(3\) can be expressed as \(3^2\) and \(3^2 = 3 \times 3 = 9\text{.}\)

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Activity 1.5.1.

Work in groups

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(a)

Draw a grid of squares like the one shown below.

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(b)

Start by counting all the smallest squares \((1 \times 1)\) in the grid.

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(c)

Then, identify and count all the larger squares, such as \(2 \times 2, \,\,\) \(3 \times 3\text{,}\) and so on, until you reach the largest square that fits within the grid.

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(d)

Write down the number of squares for each size \((e.g., 1 \times 1, \,\, 2 \times 2, ...) \) in a table for better organization. Add them up to find the total number of squares in the grid.

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(e)

Discuss your findings in a small group. Compare how many squares each of you found and check for any missed squares.

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(f)

Think about the process you used to count the squares. Could there be a quicker way to calculate the total number of squares without counting each one individually?

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Example 1.5.2.

What is the square of \(17 \text{?}\)
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Solution.

The square of a number is obtained by multiplying the number by itself.

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Now, express \(17^2\) as \(17 \times 17\text{.}\)

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Then, perform multiplication.

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\(17\)
\(\times \, 17\)
\(119\)	\({\color{blue}17} \times {\color{blue}7} \)
\(+ 17\,\,\,\,\)	\({\color{blue}17} \times {\color{blue}1} \)
\(289\)

\begin{align*} 17^2 = 17 \times 17 \amp = 289\\ \amp = 289 \end{align*}

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Example 1.5.3.

The length of a square floor tile is \(22\) cm. What is the area of the tile?

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Solution.

\begin{align*} \text{Area of the square floor time} \amp = \text{ Length of the tile } \times \text{Length of the tile} \\ \amp = 22 \text{ cm } \times 22 \text{ cm } \\ \amp = 484 \text{ cm}^2 \end{align*}

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\(22\)
\(\times 22\)
\(44\)	\({\color{blue}22} \times {\color{blue}2}\)
\(+ 44\,\,\,\,\)	\({\color{blue}22} \times {\color{blue}2}\)
\(484\)

The area of the floor tile is \(484 \text{ cm}^2.\)

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Checkpoint 1.5.4.

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Checkpoint 1.5.5.

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Subsection 1.5.2 Squares of fractions

We have seen how to find squares of whole numbers Subsection 1.5.1. But what about fractions? A fraction contains of a numerator and a denominator, both of which are whole numbers. Squaring a fraction follows the same way as squaring whole numbers, in which we simply multiply both the numerator and the denominator by themselves.

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Activity 1.5.2.

Work in groups

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(a)

Wekesa, a Grade 7 student, created the table below.Wekesa, a Grade 7 student, created the table below.

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Fraction	Square of the fraction	Squared fraction
\(\left( \frac{2}{3}\right)^2\)	\(\frac{2^2}{3^2} = \frac{2 \times 2}{3 \times 3}\)	\(\frac{4}{9}\)
\(\left( \frac{7}{15}\right)^2\)
\(\left( \frac{5}{12}\right)^2\)
\(\left( \frac{4}{9}\right)^2\)
\(\left( \frac{11}{14}\right)^2\)
\(\left( \frac{1}{3}\right)^2\)

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(b)

Help Wekesa to complete filling in the table.

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(c)

Share your work with other learners in class

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Example 1.5.6.

Work out the square of:

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\(\displaystyle \frac{2}{5}\)
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Solution.

\(\left( \frac{2}{5} \right)^2\)
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To square a fraction, square both the numerator (top number) and the denominator (bottom number). \(\left( \frac{2}{5} \right)^2 = \frac{2^2}{5^2}\)
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Square the numerator: \(2^2 = 2 \times 2 = 4\)
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Square the denominator: \(5^2 = 5 \times 5 = 25\)
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Combine the squared numerator and denominator to form the new fraction.
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\(\left( \frac{2}{5} \right)^2 = \frac{2^2}{5^2} = \frac{4}{25}\)
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Example 1.5.7.

Work out the following:

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\(\displaystyle \left(2 \frac{2}{7}\right)^2\)
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Solution.

\(\left(2 \frac{2}{7}\right)^2\)

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Convert the mixed fraction to an improper fraction.

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\begin{equation*} 2\frac{2}{7} = \frac{(7 \times 2) + 2}{7}= \frac{16}{7} \end{equation*}

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Find the square of the improper fraction by squaring both the numerator and denominator.

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\begin{align*} \left(\frac{16}{7}\right)^2 = \frac{16^2}{7^2} \amp = \frac{16 \times 16}{7 \times 7}\\ \amp = \frac{256}{49} \end{align*}

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Convert the improper fraction \(\frac{256}{49}\) to a mixed fraction.

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\begin{align*} \frac{256}{49} \amp = 5 \frac{11}{49} \\ \\ \amp = 5 \frac{11}{49} \end{align*}

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Checkpoint 1.5.8.

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Checkpoint 1.5.9.

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Subsection 1.5.3 Squares of decimals

Squaring a decimal number means multiplying the decimal number by itself. To do this, multiply the number by itself, then count the total decimal places in the original numbers and apply the same number of decimal places in the final product. Alternatively, we can convert the decimal into a fraction, square the fraction, and then simplify it back to a decimal number.

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The square of a decimal number should have double the number of decimal places as the original number.

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Activity 1.5.3.

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(a)

Drag both sliders so they have the same value.

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(b)

click the MULTIPLY button.

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(c)

What did you observe from the shaded region in the three squares.

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(d)

Click START AGAIN button and repeate the process with different decimal values

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Figure 1.5.10.

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Example 1.5.11.

What is the square of: \(1.21\text{?}\)
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Solution.

\(1.21^2\)
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There are several approaches that one can use to find the square of \(1.21\)
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In this approach, we convert \(1.21\) to a fraction.
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\(1.21 = \frac{121}{100}\)
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Then, we square the fraction \(\frac{121}{100}\text{,}\) multiply the numerator by itself and the denominator by itself.
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\(\frac{121}{100} \times \frac{121}{100} = \frac{121 \times 121}{100 \times 100} = \frac{14641}{10000}\)
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We finally convert the fraction \(\frac{14641}{10000}\) to a decimal by dividing the numerator by the denominator.
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\(\frac{14641}{10000} = 1.4641\)
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Example 1.5.12.

What is the value of \(2.35^2\text{?}\)

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Solution.

\(2.35^2\)

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To find \(2.35^2\) we need to calculate \(2.35 \times 2.35\)

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Convert \(2.35\) to a whole number by multiplying it by 100.

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\(2.35 \times 100 = 235\)

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Then, we have: \(235 \times 235 \)

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\(235\)
\(\times 235\)
\(1175\)	\({\color{blue}235} \times {\color{blue}5}\)
\(705\)	\({\color{blue}235} \times {\color{blue}3}\)
\(+ 470\)	\({\color{blue}235} \times {\color{blue}2}\)
\(55\,225\)

Since we initially removed two decimal places from each number \(2.35\) (multiplying by 100 for both), we must divide the final result by \(100^2 = 10000\) to restore the decimal:

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\begin{align*} 55225 \div 10000 = \frac{55225}{10000} \amp = 5.5225 \\ \amp = 5.5225 \end{align*}

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Checkpoint 1.5.13.

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Subsection 1.5.4 Squares roots of whole numbers

The square root of a number is the inverse operation of squaring a number. The square of a number is the value that is obtained when we multiply the number by itself, while the square root of a number is obtained by finding a number that when squared gives the original number.

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We represent the square root of a number using this symbol \(\sqrt{}\text{.}\) For example, square root of 4, we represent it as \(\sqrt{4}.\)

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To find the square root of a number, we just see by squaring which number would give the original number. It is very easy to find the square root of a number that is a perfect square. Perfect squares are those positive numbers that can be expressed as the product of a number by itself.

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Activity 1.5.4.

Work in groups

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(a)

Draw the following grid pattern in your exercise books or on graph paper, as the one shown below.

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The image displays a three by three grid composed of square sections. — Figure 1.5.14. Grid 1
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The image displays a four by four grid composed of square sections. — Figure 1.5.15. Grid 2
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The image displays a five by five grid composed of square sections. — Figure 1.5.16. Grid 3
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(b)

Count all the small squares in each grid.

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(c)

Count all squares along one side of each grid to identify its side length.

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(d)

Analyze the relationship between the side length and the total number of squares in the grid. What pattern do you observe?

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Example 1.5.17.

Find the square root of \(256\text{.}\)

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Solution.

Express \(256\) as a product of its prime factors and pair up similar factors together.

\begin{align*} 256 \amp = {\color{Green} 2} \times {\color{Green} 2} \times {\color{blue} 2} \times {\color{blue} 2} \times {\color{red} 2} \times {\color{red} 2} \times 2 \times 2\\ \\ \sqrt{324} \amp = \sqrt{{\color{Green} 2} \times {\color{Green} 2} \times {\color{blue} 2} \times {\color{blue} 2} \times {\color{red} 2} \times {\color{red} 2} \times 2 \times 2} \end{align*}

select one factor from each pair and multiply the selected factors.

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\begin{align*} \amp = {\color{Green} 2} \times {\color{blue} 2} \times {\color{red} 2} \times 2\\ \amp = 16 \end{align*}

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Checkpoint 1.5.18.

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Subsection 1.5.5 Square roots of fractions

Now that we understand how to find the square of a fraction, as discussed in Subsection 1.5.2, let’s explore how to calculate the square root of a given fraction. To do this, we simply find the square roots of the numerator and the denominator separately. The last step will be dividing the square root of the numerator by the square root of the denominator.

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Activity 1.5.5.

Work in groups

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(a)

Draw the following grids as shown below.

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This image shows a two by two grid square with one small squares colored — Figure 1.5.19. Grid 1
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This image shows a four by four grid square with nine small squares colored — Figure 1.5.20. Grid 2
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This image shows a three by three grid with four small squares colored — Figure 1.5.21. Grid 3
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(b)

Count and record the total number of small squares in each shaded region of the grid.

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(c)

Count and record the total number of small squares in each grid.

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(d)

Form a fraction using the total number of shaded squares as the numerator and the total number of squares in the grid as the denominator.

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(e)

Identify the side length of each square grid and count the number of small squares along one side.

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(f)

Identify the side length of the shaded region and count the number of small squares along one side.

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(g)

Form a fraction using the total number of squares gotten from step \(\textbf{(e)}\) as the numerator and total number of squares gotten from step \(\textbf{(f)}\) as the denominator.

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(h)

Compare the two fractions obtained in step \(\textbf{(d)}\) and \(\textbf{(g)}\text{.}\) What relationship do you observe between them?

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(i)

Share your group discussion work, with other learners in class

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To find the square root of a fraction, work out the square root of the numerator and square root of the denominator separately.

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Example 1.5.22.

Find the square root of \(\frac{36}{400}\text{.}\)

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Solution.

To find the square root of \(\frac{36}{400}\text{,}\) take the square root of numerator and denominator separately.

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\begin{equation*} \sqrt{\frac{36}{400}} = \frac{\sqrt{36}}{\sqrt{400}} \end{equation*}

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Express the numerator and denominator in terms of its prime factor pairs.

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\begin{equation*} \frac{\sqrt{36}}{\sqrt{400}} = \frac{\sqrt{{\color{blue}2} \times {\color{blue}2} \times {\color{red} 3} \times {\color{red}3}}}{\sqrt{{\color{blue}2} \times {\color{blue}2} \times {\color{red}2} \times {\color{red}2} \times 5 \times 5}} \end{equation*}

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Take one factor from each pair, and multiply.

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\begin{align*} \frac{{\color{blue}2}}{{\color{blue}2} \times {\color{red}2} \times 5} \amp = \frac{6}{20}\\ \\ \amp = \frac{6}{20} \end{align*}

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Example 1.5.23.

Work out: \(\sqrt{28\frac{4}{9}}\)

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Solution.

To find the \(\sqrt{28\frac{4}{9}}\text{,}\) first convert \(28\frac{4}{9}\) into improper fraction.

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\begin{equation*} 28 \frac{4}{9} = \frac{(9 \times 28)+4}{9} = \frac{256}{9} \end{equation*}

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Now, \(\sqrt{28\frac{4}{9}} = \sqrt{\frac{256}{9}} = \frac{\sqrt{256}}{\sqrt{9}}\)

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Express the numerator and denominator in terms of its prime factor pairs.

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\begin{equation*} \frac{\sqrt{256}}{\sqrt{9}} = \frac{\sqrt{{\color{blue} 2} \times {\color{blue} 2} \times {\color{red} 2} \times {\color{red} 2} \times {\color{green} 2} \times {\color{green} 2} \times 2 \times 2 }}{\sqrt{{\color{blue}3} \times {\color{blue}3}}} \end{equation*}

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Take one factor from each pair, and multiply.

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\begin{align*} \frac{{\color{blue} 2} \times {\color{red} 2} \times {\color{green} 2} \times 2}{{\color{blue}3} } \amp = \frac{16}{3}\\ \\ \amp = \frac{16}{3} \end{align*}

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Checkpoint 1.5.24.

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Subsection 1.5.6 Square roots of decimals

Finding the square root of a decimal number involves similar method of finding the square root of whole numbers, but with careful attention to the placement of the decimal point.

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We can express the given decimal number as a fraction and then find the square roots of the numerator and denominator separately. We then simplify the resulting fraction and convert it back into a decimal number.

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Activity 1.5.6.

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(a)

Drag both sliders so they have the same value.

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(b)

click the MULTIPLY button.

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(c)

What did you observe from the shaded region in the three squares.

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(d)

Click START AGAIN button and repeate the process with different decimal values

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Figure 1.5.25.

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Example 1.5.26.

Work out: \(\sqrt{0.81}\)
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Solution.

\(\sqrt{0.81}\)
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Since \(0.81\) can be written as the fraction \(\frac{81}{100}\text{,}\) we take the square root of both the numerator and the denominator separately.
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\begin{equation*} \sqrt{0.81} = \sqrt{\frac{81}{100}} = \frac{\sqrt{81}}{\sqrt{100}} \end{equation*}

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Here, we express \(81\) and \(100\) in terms of their prime factors to simplify the square root calculation.
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\begin{equation*} \frac{\sqrt{81}}{\sqrt{100}} = \frac{\sqrt{{\color{red}3} \times {\color{red}3} \times {\color{blue}3} \times {\color{blue}3}}}{\sqrt{{\color{red}2} \times {\color{red}2} \times {\color{blue}5} \times {\color{blue}5}}} \end{equation*}

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Select one factor from each pair and multiply the selected factors.
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\begin{equation*} \frac{{\color{red}3} \times {\color{blue}3} }{{\color{red}2} \times {\color{blue}5}} = \frac{9}{10} \end{equation*}

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Divide the results:
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\begin{equation*} \frac{9}{10} = 0.9 \end{equation*}

\begin{equation*} \quad = 0.9 \end{equation*}

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Checkpoint 1.5.27.

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